UDC 517.917

INVARIANTS OF LINEAR REPRESENTATIONS OF THE ROTATION GROUP OF THE PLANE AND THEIR APPLICATIONS IN THE QUALITATIVE THEORY OF DIFFERENTIAL EQUATIONS*)

K. S. SIBIRSKII

§ 3. CONDITIONS FOR SYMMETRY OF THE DIRECTION FIELD OF A DIFFERENTIAL EQUATION

1. Axes of symmetry.

By an axis of symmetry of the differential equation

\[ \frac{dy}{dx} = -\frac{\sum_{j+l\in A} c_{jl} x^j y^l} {\sum_{j+l\in A} b_{jl} x^j y^l} \tag{3.1} \]

we shall mean an axis of symmetry, passing through the origin of coordinates, of the direction field of this equation.

System (2.5) defines in the phase plane \(XOY\) a family of orthogonal trajectories for the integral curves of equation (3.1). Therefore the axes of symmetry of the direction field of equation (3.1) coincide with the axes of symmetry of the direction field defined by system (2.5) in the phase plane \(XOY\).

We note, moreover, that the integral curves of equation (3.1) coincide with the trajectories of the system

\[ \frac{dx}{dt}=\pm \sum_{j+l\in A} b_{jl}x^j y^l,\qquad \frac{dy}{dt}=\mp \sum_{j+l\in A} c_{jl}x^j y^l, \tag{3.2} \]

which is equivalent to the equation

\[ \pm i\,\frac{dw}{dt}=F(w). \]

Comparing this equation with (2.8), one may conclude that systems (2.5) and (3.2) have the same invariants.

We shall call the right-hand side of equation (3.1) irreducible if in it \(P(x,y)\) and \(Q(x,y)\) are relatively prime polynomials, or, equivalently, the function \(F(w)\) is a polynomial in \(\overline{w}\) and \(w\) having no real factors of nonzero degree.

Denote by \(Z^*\) the set of points of the space \(Z\) for which the right-hand side of equation (3.1) is irreducible.

Let us find the conditions under which the axis \(OX\) is an axis of symmetry of equation (3.1). Namely, we shall show that the following holds.

Lemma 3.1. In order that the axis \(OX\) be an axis of symmetry of equation (3.1), it is sufficient, and for \(z\in Z^*\) necessary, that the condition

\[ \overline{z}=\pm z. \]

* For the beginning of the article see the journal Differential Equations, No. 6, 1966.

Proof. In order that the \(OX\) axis be an axis of symmetry of equation (3.1), it is necessary and sufficient that, in the domain (2.4), the identity

\[ \frac{Q(x,-y)}{P(x,-y)} \equiv -\frac{Q(x,y)}{P(x,y)} . \tag{3.3} \]

hold.

Suppose now that the condition \(\bar z=\pm z\) is satisfied. Then \(z_{jl}=\pm \bar z_{jl}\) and \(F(\bar w)=\pm \overline{F(w)}\). Hence
\(Q(x,-y)+iP(x,-y)\equiv \pm (Q(x,y)-iP(x,y))\), and condition (3.3) holds. Thus the sufficiency of the condition \(\bar z=\pm z\) for the \(OX\) axis to be an axis of symmetry of equation (3.1) has been proved.

Assume now that \(z\in Z^*\) and that condition (3.3) is satisfied. Then
\(Q(x,-y)P(x,y)\equiv -P(x,-y)Q(x,y)\). Since \(P\) and \(Q\) are relatively prime polynomials, it follows that there exists a real number \(k\ne0\) such that
\(P(x,-y)\equiv kP(x,y)\), \(Q(x,-y)\equiv -kQ(x,y)\). In this case
\(F(\bar w)=k\overline{F(w)}\), whence \(\bar z=kz\). Then \(z=k\bar z\), and consequently \(z=k^2z\). Since \(z\ne0\), it follows that \(k^2=1\), and \(k=\pm1\). Thus \(\bar z=\pm z\), and the lemma is proved.

The conditions \(\bar z=z\) and \(\bar z=-z\), obviously, cannot hold simultaneously, since \(z\ne0\).

Now rotating the coordinate axes in the plane \(XOY\) through an angle \(\varphi\) \((0\le \varphi<2\pi)\) according to formula (2.15), and taking into account that \(\bar U_\varphi z=U_{-\varphi}\bar z\), we arrive at the following lemma.

Lemma 3.2. In order that the line \(x\sin\varphi-y\cos\varphi=0\) be an axis of symmetry of the differential equation (3.1), it is sufficient, and for \(z\in Z^*\) necessary, that the condition

\[ U_{-\varphi}\bar z=\pm U_\varphi z \tag{3.4} \]

be satisfied.

Both of these conditions cannot be satisfied for one and the same \(\varphi\), since then we would have \(U_\varphi z=0\), i.e. \(z=0\). The following holds.

Theorem 3.1. For an axis of symmetry of equation (3.1) to exist, it is sufficient, and for \(z\in Z^*\) necessary, that at least one of the conditions

\[ J(\bar z)=J(z), \tag{3.5} \]

\[ J(\bar z)=J(-z) \tag{3.6} \]

hold simultaneously for all algebraic invariants \(J\) of the group \(U\) on \(P\), or, equivalently, for some complete system of algebraic invariants of the group \(U\).

Proof. For the existence of an axis of symmetry of equation (3.1), by Lemma 3.2, it is sufficient, and for \(z\in Z^*\) necessary, that equation (3.4) be solvable with respect to \(\varphi\). By Corollary 1.1, equation (3.4) has a solution if and only if at least one of the two conditions (3.5) and (3.6) is satisfied. Here and everywhere below, when the satisfaction of condition (3.5) or (3.6) is mentioned, its satisfaction is always meant for all algebraic invariants \(J\) of the group \(U\) on \(P\), or, equivalently, for some complete system of algebraic invariants of the group \(U\).

Theorem 3.2. For \(z\in Z^*\), the number of axes of symmetry of equation (3.1) is equal to \([U(z)]\), if one of conditions (3.5) and (3.6) is satisfied, and is equal to \(2[U(z)]\), if both of these conditions are satisfied.

Proof. Let \(z\in Z^*\). Then, by Lemma 3.2, all axes of symmetry of equation (3.1) are determined from equation (3.4).

Let us now suppose that condition (3.5) is satisfied. Then, on the basis of Corollary 1.1, the equation \(U_{-\varphi}\bar z=U_\varphi z\) determines in the plane \(XOY\) exactly \([U(z)]\) axes of symmetry.

An analogous assertion holds for the case when condition (3.6) is satisfied.

Since, as was noted above, the equations \(U_{-\varphi}\bar z=U_\varphi z\) and \(U_{-\varphi}\bar z=-U_\varphi z\) cannot have common solutions, Theorem 3.2 is proved.

Remark 3.1. Let \(R(z)=R\). Suppose that, of the conditions (3.5) and (3.6), only one is satisfied. Then among the invariants \(J\) there must be invariants of odd degree, and therefore, according to Lemma 1.5, either \(\varkappa=0\), or among the coordinates of the vector \(\varkappa\) there are both even and odd ones. If \(\varkappa=0\), i.e. \([U]=\infty\), then the condition \(U_{-\varphi}\bar z=\pm U_\varphi z\) is satisfied by all straight lines passing through the origin. If, however, \(\varkappa\ne0\), then by Lemma 1.6, among the positive elements \((\gamma,\delta)\in\Delta_l\times\Delta_l\) satisfying equation (1.14), there will be both elements with even \(\|\gamma+\delta\|_l\) and elements with odd \(\|\gamma+\delta\|_l\).

If, in addition, condition (3.5) is satisfied, then the equation \(U_{-\varphi}\bar z=U_\varphi z\) is equivalent to the equation

\[ (\bar z;\gamma,\delta)=(z;\gamma,\delta)\exp(2[U]i\varphi). \tag{3.7} \]

If condition (3.6) is satisfied, then the equation \(U_{-\varphi}\bar z=-U_\varphi z\) is equivalent to the equation obtained from (3.7) by replacing \(z\) by \(-z\) on its right-hand side, i.e. it is equivalent to equation (3.7) if \(\|\gamma+\delta\|_l\) is even, and to the equation

\[ (\bar z;\gamma,\delta)=-(z;\gamma,\delta)\exp(2[U]i\varphi), \tag{3.8} \]

if \(\|\gamma+\delta\|_l\) is odd.

Let us now suppose that both conditions (3.5) and (3.6) are satisfied. For this it is necessary and sufficient that at least one of them be satisfied and that all invariants \(J\) have even degrees, i.e. (see Lemma 1.5) all coordinates of the vector \(\varkappa\) be odd. In this case, according to Lemma 1.6, \(\|\gamma+\delta\|_l\) is always odd. Then the equations \(U_{-\varphi}\bar z=U_\varphi z\) and \(U_{-\varphi}\bar z=-U_\varphi z\) are equivalent, respectively, to equations (3.7) and (3.8).

If \(R(z)\subset R\), then all the preceding arguments can be carried out by considering the group \(U\) and equations (1.14), (3.7), and (3.8) on the subspace \(Z(z)\).

We note that equations (3.7) and (3.8) can be rewritten respectively in the form

\[ \operatorname{Im}\{(z;\gamma,\delta)\exp([U]\,i\varphi)\}=0 \quad\text{and}\quad \operatorname{Re}\{(z;\gamma,\delta)\exp([U]\,i\varphi)\}=0. \tag{3.9} \]

From Theorem 3.2 it follows that

Corollary 3.1. For \(q<\infty\), either the number of axes of symmetry of equation (3.1) is infinite, or it does not exceed \(2q+2\).

To prove this assertion, one should first note that, according to (2.17), all components of the characteristic vector \(\zeta\) of the group \(U\) satisfy the inequality \(|\zeta_r|\le q+1\), and therefore either \([U(z)]=\infty\), or \([U(z)]\le q+1\). Secondly, one must take into account that, when the numerator and denominator of the right-hand side of equation (3.1) are reduced by a possible common factor, the number of axes of symmetry cannot decrease.

The equation

\[ \frac{dy}{dx}=-\frac{\operatorname{Re}\{(x-iy)^q\}}{\operatorname{Im}\{(x-iy)^q\}}, \]

which has \(2q+2\) axes of symmetry, shows the sharpness of the estimate given here for all \(q\).

2. Examples.

According to the results at the end of the preceding paragraph, conditions (3.5) and (3.6) for the differential equation

\[ \frac{dy}{dx}=-\,\frac{c_{10}x+c_{01}y}{b_{10}x+b_{01}y} \]

will respectively take the form \(\bar z_{01}=z_{01}\) and \(\bar z_{01}=-z_{01}\).

Assuming the right-hand side of this equation to be irreducible

\[ \left(\left|\begin{matrix} c_{10} & c_{01}\\ b_{10} & b_{01} \end{matrix}\right|\ne 0\right), \]

we arrive at the conclusion that axes of symmetry will exist only in the following cases:

1) \(z_{01}=0,\ z_{10}\ne 0\). The equation has 4 axes of symmetry, determined by the equations

\[ \operatorname{Im}\,[\bar z_{10}\exp(2i\varphi)]=0 \quad\text{and}\quad \operatorname{Re}\,[\bar z_{10}\exp(2i\varphi)]=0, \]

which are easily obtained from (3.9);

2) \(z_{10}=0,\ z_{01}\ne 0\) and either \(\operatorname{Im} z_{01}=0\), or \(\operatorname{Re} z_{01}=0\). In this case all straight lines passing through the origin are axes of symmetry;

3) \(z_{10}z_{01}\ne 0\) and either \(\operatorname{Im} z_{01}=0\), or \(\operatorname{Re} z_{01}=0\). The equation has 2 axes of symmetry, determined respectively by the equation

\[ \operatorname{Im}\,[\bar z_{10}\exp(2i\varphi)]=0 \quad\text{or}\quad \operatorname{Re}\,[\bar z_{10}\exp(2i\varphi)]=0. \]

Let us consider also, as an example, the differential equation studied by L. S. Lyagina [23],

\[ \frac{dy}{dx} = -\frac{c_{20}x^2+c_{11}xy+c_{02}y^2} {b_{20}x^2+b_{11}xy+b_{02}y^2}. \tag{3.10} \]

In order to construct a fundamental basis of algebraic invariants of the group \(U\) in this case, using Theorem 1.2, we compile the following table of minimal elements \((\alpha,\beta)\) of the set of positive points \((\psi,\omega)\in \Delta_l\times\Delta_l\) satisfying equation (1.4):

These elements determine the following 13 invariants, which in the present case constitute a fundamental basis:

\[ z_{20}\bar z_{20},\quad z_{11}\bar z_{11},\quad z_{02}\bar z_{02},\quad z_{11}\bar z_{02},\quad \bar z_{11}z_{02},\quad z_{20}\bar z_{11}^{\,3},\quad z_{20}\bar z_{11}^{\,2}z_{02},\quad \bar z_{20}z_{11}z_{02}^{\,2}, \]

\[ z_{20}z_{02}^{\,3},\quad \bar z_{20}z_{11}^{\,3},\quad \bar z_{20}z_{11}^{\,2}\bar z_{02},\quad \bar z_{20}z_{11}\bar z_{02}^{\,2},\quad \bar z_{20}\bar z_{02}^{\,3}. \]

As for the basic complete system (1.12) of algebraic invariants of the group \(U\), in the present case it consists of the following 6 invariants:

\[ z_{20}\bar z_{20},\quad z_{11}\bar z_{11},\quad z_{02}\bar z_{02},\quad z_{11}\bar z_{02},\quad z_{20}\bar z_{11}^{\,3},\quad z_{20}z_{02}^{\,3}. \]

The conditions (3.5), which in this case coincide with the conditions (3.6), since all invariants have even degrees, can be written in the form

\[ \operatorname{Im}(z_{11}\bar z_{02})= \operatorname{Im}(z_{20}\bar z_{11}^{\,3})= \operatorname{Im}(z_{20}z_{02}^{\,3})=0. \tag{3.11} \]

Suppose now that the right-hand side of equation (3.10) is irreducible. Then the fulfillment of conditions (3.11) is not only sufficient but also necessary for the existence of an axis of symmetry of equation (3.10). Moreover, if conditions (3.11) are fulfilled, then equation (3.10) has 2 axes of symmetry if \(|z_{11}|+|z_{02}|\ne0\), and 6 such axes if \(z_{11}=z_{02}=0\).

These axes are determined from the equations

\[ \operatorname{Im}[z_{02}\exp(i\varphi)]=0 \quad\text{and}\quad \operatorname{Re}[z_{02}\exp(i\varphi)]=0 \quad\text{for } z_{02}\ne0, \]

\[ \operatorname{Im}[\bar z_{11}\exp(i\varphi)]=0 \quad\text{and}\quad \operatorname{Re}[\bar z_{11}\exp(i\varphi)]=0 \quad\text{for } z_{11}\ne0, \]

\[ \operatorname{Im}[\bar z_{20}\exp(3i\varphi)]=0 \quad\text{and}\quad \operatorname{Re}[\bar z_{20}\exp(3i\varphi)]=0 \quad\text{for } z_{11}=z_{02}=0, \]

which are easily obtained from (3.9).

3. Criteria for the absence of axes of symmetry. We shall establish some cases when equation (3.1) cannot have axes of symmetry distinct from the coordinate axes.

Lemma 3.3. If \(z\in Z^*\) and there exists a pair of numbers \(j\) and \(l\) and a real number \(c\) such that

\[ z_{jl}=c\bar z_{lj}\ne0, \tag{3.12} \]

then the axes of symmetry of equation (3.1) can only be coordinate axes.

Proof. On the basis of Lemma 3.1, all axes of symmetry of equation (3.1) are determined from the equalities \(U_{-\varphi}\bar z=U_\varphi z\) and \(U_{-\varphi}\bar z=-U_\varphi z\), i.e., from the equalities \(\operatorname{Im}U_\varphi z=0\) and \(\operatorname{Re}U_\varphi z=0\). In particular, either the imaginary or the real parts of the expressions

\[ z_{jl}\exp[(l-j-1)i\varphi] \quad\text{and}\quad z_{lj}\exp[(j-l-1)i\varphi] \]

must be equal to zero. Multiplying these expressions in both cases and taking (3.12) into account, we obtain that necessarily

\[ \operatorname{Im}[\exp(2i\varphi)]=\sin2\varphi=0, \]

whence

\[ \varphi=n\frac{\pi}{2}, \]

and the lemma is proved.

In particular, the conditions of Lemma 3.3 are satisfied if \(z\in Z^*\) and there exists an \(n\in A\) and a real number \(c\) such that

\[ z_{jl}=c\bar z_{lj} \quad\text{for all } j+l=n, \tag{3.13} \]

and at least one \(z_{jl}\) with \(j+l=n\) is different from zero. From (3.13) it follows that \(c=\pm1\). Conditions (3.13) with \(c=1\) (\(c=-1\)) are equivalent to the fact that \(b_{jl}=0\) (\(c_{jl}=0\)) for all \(j+l=n\), and at least one \(c_{jl}\) (\(b_{jl}\)) with \(j+l=n\) is different from zero (see formulas (2.10), (2.11), and (2.13)). Thus, we obtain that the following holds.

Theorem 3.3. If \(z\in Z^*\) and there exists an \(n\in A\) such that one of the sums

\[ \sum_{j+l=n} c_{jl}x^j y^l \quad\text{and}\quad \sum_{j+l=n} b_{jl}x^j y^l \]

identically equal to zero, while the second does not have this property, equation (3.1) cannot have any axes of symmetry other than the coordinate axes.

If \(A\) is finite and \(n=\inf A\) or \(n=\sup A\), then it is obvious that the irreducibility conditions for the right-hand side of equation (3.1) in Theorem 3.3 may be removed. The example
\[ \frac{dy}{dx}=-\frac{x-xy^3}{y-xy+y^2-xy^2+y^3-xy^3} \tag{3.14} \]
shows that in other cases irreducibility is essential.

From Theorem 3.3 one obtains, in particular, the well-known result of I. S. Kukles [6] that the equations
\[ \frac{dy}{dx}=-\frac{x+\sum^{p}_{j+l=2} c_{jl}x^j y^l}{y} \quad\text{and}\quad \frac{dy}{dx}=-\frac{x}{y+\sum^{p}_{j+l=2} b_{jl}x^j y^l} \]
cannot have any axes of symmetry (passing through the origin) other than the coordinate axes themselves, provided only that
\(\sum^{p}_{j+l=2} c_{jl}x^j y^l\not\equiv0\) and
\(\sum^{p}_{j+l=2} b_{jl}x^j y^l\not\equiv0\), respectively.

One can also indicate other criteria for the absence of axes of symmetry different from the coordinate axes. Such is, for example, the case when the vector \(z\in Z^*\) has at least one pair of nonzero coordinates \(z_{jl}\) and \(z_{rs}\), differing by a real factor, and the corresponding coordinates of the vector \(\zeta\), i.e. the numbers \(l-j-1\) and \(s-r-1\), differ by two units \((l-j-s+r=\pm2)\).

§ 4. APPLICATION TO THE PROBLEM OF CENTER AND FOCUS

1. Axes of symmetry in the case of a singular point of the second group. Consider differential equation (3.1) under the assumption that
\[ \sum_{j+l=p}(b_{jl}y+c_{jl}x)x^j y^l\ne0 \quad\text{for } x^2+y^2\ne0. \tag{4.1} \]
It is obvious that in this case the number \(p=\inf A\) must be odd.

As is known [7], when this condition is fulfilled the origin is, for equation (3.1), a singular point having no exceptional directions, and therefore it belongs to the second group of singular points, i.e. it is a center or a focus.

Condition (4.1) is equivalent to
\[ \operatorname{Re}\bigl(\overline{w}\sum_{j+l=p} v_{jl}x^j y^l\bigr)\ne0 \quad\text{for } w\ne0, \]
or, what is the same,
\[ \operatorname{Re}\bigl(\sum_{j+l=p} z_{jl}\overline{w}^{\,j+1}w^l\bigr)\ne0 \quad\text{for } w\ne0, \]
i.e.
\[ \sum_{j+l=p}\bigl(z_{jl}\overline{w}^{\,j+1}w^l+\overline{z}_{jl}w^{j+1}\overline{w}^{\,l}\bigr)\ne0 \quad\text{for } w\ne0. \]
Making here the transformation (2.15), we obtain
\[ \sum_{j+l=p}\bigl(z_{jl}(\varphi)\overline{w}_1^{\,j+1}w_1^l +\overline{z}_{jl}(\varphi)w_1^{j+1}\overline{w}_1^{\,l}\bigr)\ne0 \quad\text{for } w_1\ne0, \tag{4.2} \]
where \(z_{jl}(\varphi)\) are determined by formulas (2.16). Hence, in particular, it is seen that condition (4.1) is invariant with respect to rotations of the coordinate axes.

There holds

Lemma 4.1. Under condition (4.1), the equation \(U_{-\varphi}\overline{z}=-U_{\varphi}z\) has no solutions, and condition (3.6) cannot be satisfied.

Indeed, if condition (3.6) is satisfied, and only in this case (see Corollary 1.1), there exists a number \(\varphi\in[0,2\pi)\) satisfying the equation
\(U_{-\varphi}\overline{z}=-U_{\varphi}z\). For this \(\varphi\) we have
\(\overline{z}_{jl}(\varphi)=-z_{jl}(\varphi)\), and then the sum appearing in (4.2) vanishes for \(w_1=\overline{w}_1\ne0\); consequently, condition (4.1) cannot be satisfied. The lemma is proved.

Denote by \(\widetilde Z\) the set of points of the space \(Z\) for which the right-hand side of equation (3.1) is irreducible and condition (4.1) is satisfied.

From Lemmas 3.2 and 4.1 it follows

Lemma 4.2. In order that the line \(x\sin\varphi-y\cos\varphi=0\) be an axis of symmetry of the differential equation (3.1), it is sufficient, and for \(z\in\widetilde Z\) necessary, that the conditions

\[ \operatorname{Im} z_{jl}(\varphi)=0\quad (j+l\in A) \tag{4.3} \]

be satisfied.

It is not difficult to see that conditions (4.3) coincide with conditions (4) of Theorem 1, stated without proof in [13].

Theorem 4.1. For the existence of an axis of symmetry of equation (3.1) it is sufficient, and for \(z\in\widetilde Z\) necessary, that all algebraic invariants of the group \(U\) over \(D\), or, equivalently, all invariants of some complete system of algebraic invariants of the group \(U\) over \(D\), have real values at the point \(z\).

This theorem follows directly from Theorem 3.1 and Lemma 4.1, if one takes into account that for any algebraic invariant \(J\) of the group \(U\) over \(D\) the relation \(J(\bar z)=\overline{J(z)}\) holds, and therefore condition (3.5) is equivalent to

\[ \operatorname{Im} J(z)=0 \tag{4.4} \]

for all algebraic invariants of the group \(U\) over \(D\), or, what is the same, for some complete system of algebraic invariants of the group \(U\) over \(D\).

From Lemma 4.1, Theorem 3.2, and Remark 3.1 it follows

Theorem 4.2. If \(z\in\widetilde Z\) and condition (4.4) is satisfied, then the number of axes of symmetry of equation (3.1) is equal to the characteristic number \([U(z)]\) of the group \(U(z)\) (and, consequently, does not exceed \(q+1\), if it is finite).

Remark 4.1. Let \(z\in\widetilde Z\) and let condition (4.4) be satisfied. Then, for \([U(z)]=\infty\), the axes of symmetry of equation (3.1) are all lines passing through the origin. For \([U(z)]<\infty\) and \(R(z)=R\), all axes of symmetry are determined from the equation

\[ \operatorname{Im}\bigl[(z;\gamma,\delta)\exp([U]\,i\varphi)\bigr]=0, \tag{4.5} \]

in which the positive element \((\gamma,\delta)\in\Delta_l\times\Delta_l\) satisfies condition (1.14). If, however, \([U(z)]<\infty\) and \(R(z)\subset R\), then equations (1.14) and (4.5) must be considered on the subspace \(Z(z)\).

Suppose now that equalities (4.4) hold and condition (4.1), or some other condition ensuring that the origin belongs to the second group of singular points, is satisfied. Then, on the one hand, the origin is for equation (3.1) a singular point of the second group (i.e., a center or a focus), while, on the other hand, according to Theorem 4.1, the integral curves of this equation have an axis of symmetry passing through the origin. In this case the origin cannot be a focus. Thus, the following holds.

Theorem 4.3. If condition (4.1) or other conditions guaranteeing that the origin belongs to the second group of singular points are satisfied, the equalities (4.4) ensure the existence of a center.

In particular, conditions (4.4) are satisfied when

\[ \bar z_{jl}=c^{\,l-j-1}z_{jl}\quad (j+l\in A), \tag{4.6} \]

where \(c=1,-1,i\), or \(-i\), which is equivalent to the fulfillment of the known conditions for the existence of a center [8—11], under which the integral curves are sym-

metric with respect to the coordinate axes or the bisectors of the coordinate angles.

Indeed, from (4.6), for \(c=1\) we obtain that \(\operatorname{Im} z_{n-j,j}=B_j^{(n)}=0\); and this, as is seen from formulas (2.14) and (2.11), takes place if and only if all \(a_{jl}=0\), i.e., \(b_{jl}=0\) for even \(l\), and \(c_{jl}=0\) for odd \(l\).

For \(c=-1\), the equalities (4.6) give \(\operatorname{Im} z_{n-j,j}=B_j^{(n)}=0\) for odd \(n\) and \(\operatorname{Re} z_{n-j,j}=C_j^{(n)}=0\) for even \(n\). These conditions are equivalent to the fact that \(a_{jl}=0\) for odd \(j+l\) and \(a_{jl}^{*}=0\) for even \(j+l\), i.e., \(b_{jl}=0\) for odd \(j\), and \(c_{jl}=0\) for even \(j\).

As is seen from formulas (2.10), (2.11), and (2.13), the equalities (4.6) for \(c=i\) are equivalent to the relations \(v_{jl}=i\bar v_{lj}\), i.e., \(b_{jl}=c_{lj}\), and for \(c=-i\), to the relations \(v_{jl}=(-1)^{j+l+1}i\bar v_{lj}\), i.e., \(b_{jl}=(-1)^{j+l+1}c_{lj}\).

Thus, from (4.4) we have obtained, as a special case, all sufficient conditions for the existence of a center established in the work of M. I. Al’mukhamedov [11].

Let us note that the most thoroughly investigated case in which condition (4.1) is satisfied is represented by the equation

\[ \frac{dy}{dx}=-\frac{x+\sum_{j+l\in A'}c_{jl}x^j y^l}{y+\sum_{j+l\in A'}b_{jl}x^j y^l}, \tag{4.7} \]

in which \(A'\) is some set of distinct natural numbers not containing one. For this equation, conditions (4.4) are equivalent to the corresponding conditions for equation (3.1) with \(A=A'\).

Indeed, for equation (4.7) \(z_{10}=0,\ z_{01}=2\). If, in addition, some invariant of the basic basis contains \(z_{10}\) or \(\bar z_{10}\), then it is equal to zero and, consequently, has a real value. Moreover, the only invariants of the basic basis containing \(z_{01}\) or \(\bar z_{01}\) will be precisely \(z_{01}\) and \(\bar z_{01}\). Both of them in the present case also have real values.

In connection with the above, as conditions (4.4) for equation (4.7) with \(A'=\{2\}\), one may, for example, take conditions (3.11). Let us note that in the case of reducibility of the right-hand side of equation (4.7) for \(A'=\{2\}\) (which occurs if and only if \(c_{02}=b_{20}=0,\ c_{20}=b_{11}\), and \(c_{11}=b_{02}\)), conditions (3.11) are always satisfied, since then \(z_{20}=0\), and \(z_{02}=z_{11}\). Therefore conditions (3.11) for equation (4.7) with \(A'=\{2\}\) are not only sufficient, but also necessary for the existence of an axis of symmetry of this equation, regardless of whether its right-hand side is irreducible or not.

Equation (3.14) with a reducible right-hand side may serve as an example of an equation having the unique axis of symmetry \(y=x\), for which, however, conditions (4.4) are not satisfied.

Let us note that, discarding from the basic complete system invariants of the form \(z_{jl}\bar z_{jl}\) \((j+l\in A,\ l-j-1\ne0)\), which assume only real values, and denoting by \(J^{*}(z)\) any of the remaining \(\binom{\tilde N}{2}+N'\) invariants, the system of conditions (4.4) can be written in the form of a system of independent conditions

\[ \operatorname{Im} J^{*}(z)=0. \tag{4.8} \]

Thus, for example, for equation (4.7) with \(A'=\{2,3\}\) these conditions are written in the form

\[ \operatorname{Im}(z_{11}\bar z_{02})=\operatorname{Im}(z_{20}\bar z_{11}^{\,3})=\operatorname{Im}(z_{20}^{\,3}\bar z_{02})=\operatorname{Im}(z_{20}^{\,4}\bar z_{30}^{\,3})=\operatorname{Im}(z_{20}^{\,2}\bar z_{21}^{\,3})= \]

\[ \begin{aligned} &= \operatorname{Im}(z_{20}^2 z_{03}^3)=\operatorname{Im}(z_{11}^4 \bar z_{30}) =\operatorname{Im}(z_{11}^2 \bar z_{21}) =\operatorname{Im}(z_{11}^2 z_{03}) =\operatorname{Im}(z_{02}^4 \bar z_{30})=\\ &=\operatorname{Im}(z_{02}^2 z_{21}) =\operatorname{Im}(z_{02}^2 \bar z_{03}) =\operatorname{Im}(z_{30}^{-2} z_{21}) =\operatorname{Im}(z_{30}^2 z_{03})=\\ &=\operatorname{Im}(z_{21}z_{03})=\operatorname{Im} z_{12}=0 . \end{aligned} \tag{4.9} \]

If system (4.8) is considered not on the whole space \(Z\), but on its individual subsets, then the number of independent conditions in (4.8) may be substantially reduced. Thus, under the assumption that \(z_{11}\ne 0\), in system (4.9) only 6 independent conditions remain: \(\operatorname{Im}z_{12}=0\) and 5 conditions containing \(z_{11}\).

If \(\dim R(z)<\infty\), then, as follows from the results of item 4 of § 1, there exist \(\dim R(z)\), for \([U(z)]=\infty\), and \(\dim R(z)-1\), for \([U(z)]<\infty\), homogeneous algebraic invariants whose imaginary parts being equal to zero entails the fulfillment of conditions (4.8).

2. Simple sufficient conditions for a center. The generally known sufficient conditions for a center for equation (3.1) under condition (4.1) give the system of equalities

\[ l z_{j-1,l}=j\bar z_{l-1,j}\quad (j+l-1\in A), \tag{4.10} \]

when fulfilled, (3.1) is an equation in total differentials. Indeed, the condition \(Q_y'=P_x'\) is equivalent to \(i(F+\bar F)'_y=(F-\bar F)'_x\), or, what is the same,

\[ i^2(F+\bar F)'_w-i^2(F+\bar F)'_{\bar w} =(F-\bar F)'_w+(F-\bar F)'_{\bar w}, \]

i.e. \(F_w'=\bar F_{\bar w}'\). Taking (2.9) into account, from this we obtain conditions (4.10). At the same time, by virtue of the symmetry of formulas (4.10), it is sufficient to require the fulfillment of these conditions for \(l>j\).

Introducing the notation \(z'_{j-1,l}\equiv lz_{j-1,l}-j\bar z_{l-1,j}\) for \(l>j\), the conditions of total differential (4.10) can be rewritten in the form

\[ z'_{jl}=0\quad (j+l\in A,\ l>j+1),\qquad \operatorname{Im}z_{j,j+1}=0\quad (2j+1\in A). \tag{4.11} \]

Let us denote by \(z'\) the vector of the space \(Z\) with coordinates \(z_{jl}\) for \(l\le j+1\) and \(z'_{jl}\) for \(l>j+1\). Then a one-to-one correspondence is established between the vectors \(z\) and \(z'\) from \(Z\). It is not hard to see that, in this case, \(U_\varphi z'=(U_\varphi z)'\) (see (2.16)) and that \(J(z)\) is an algebraic invariant of the group \(U\) over \(P\) if and only if \(J_0(z)\equiv J(z')\) is such an invariant. Indeed, if \(J(z)\) is an algebraic invariant of \(U\) over \(P\), then

\[ J_0(U_\varphi z)=J((U_\varphi z)')=J(U_\varphi z')=J(z')=J_0(z), \]

and if \(J_0(z)\) is an algebraic invariant of \(U\) over \(P\), then

\[ J(U_\varphi z')=J((U_\varphi z)')=J_0(U_\varphi z)=J_0(z)=J(z'). \]

It is therefore clear that in conditions (4.4) and (4.8) one may replace \(z\) by \(z'\), and equivalent systems of conditions will be obtained. In particular, system (4.8) is equivalent to the system

\[ \operatorname{Im}J^*(z')=0 . \tag{4.12} \]

Among the \(\binom{\tilde N}{2}+N'\) conditions (4.12) there occur, in particular, \(N'\) conditions

\[ \operatorname{Im}z'_{j,j+1}=0\quad (2j+1\in A). \tag{4.13} \]

In questions connected with finding conditions for the existence of a center, it is sufficient to restrict oneself to the study of system (4.12) at those points \(z'\in Z\),

for which at least one of the conditions (4.11) is not fulfilled, i.e. (in view of (4.13)) at least one of the coordinates \(z'_{jl}\) of the vector \(z'\), for which \(l>j+1\), is nonzero. We denote by \(Z'\) the set of such vectors \(z'\in Z\).

The conditions (4.12), being independent on the whole space \(Z\), may turn out to be dependent on the set \(Z'\). There exists, however, such a subset of invariants \(\{J^{**}(z)\}\) of the set \(\{J^*(z)\}\) that the system (4.12) is equivalent on \(Z'\) to the system

\[ \operatorname{Im} J^{**}(z')=0 \tag{4.14} \]

and is not equivalent to this system if at least one invariant is removed from the set \(\{J^{**}(z)\}\).

The conditions (4.14), together with (4.1), are sufficient for the existence, at the origin of coordinates, of a center of equation (3.1). They are always satisfied when the conditions (4.4) are satisfied, but, generally speaking, are simpler than the latter.

A simple calculation shows, for example, that for \(A'=\{2\}\) the conditions (4.14) can be written in the form

\[ \operatorname{Im}(z_{11}z'_{02})=\operatorname{Im}(z_{20}z_{02}^{\prime 3})=0, \tag{4.15} \]

for \(A'=\{3\}\), in the form

\[ \operatorname{Im}z_{12}=\operatorname{Im}(z_{21}z'_{03})=\operatorname{Im}(z_{30}z_{03}^{\prime 2})=0, \tag{4.16} \]

and for \(A'=\{2,3\}\), in the form

\[ \begin{aligned} &\operatorname{Im}(z_{11}z'_{02})=\operatorname{Im}(z_{20}z_{11}^{-3}) =\operatorname{Im}(z_{20}z_{02}^{\prime 3}) =\operatorname{Im}(z_{20}^{2}z_{03}^{\prime 3})= \\ &=\operatorname{Im}(z_{11}^{2}z'_{03})=\operatorname{Im}(z_{02}^{\prime 4}z_{30}) =\operatorname{Im}(z_{02}^{\prime 2}z_{21}) =\operatorname{Im}(z_{02}^{\prime 2}\overline{z'_{03}})= \\ &=\operatorname{Im}(z_{30}z_{03}^{\prime 2}) =\operatorname{Im}(z_{21}z'_{03}) =\operatorname{Im}z_{12}=0. \end{aligned} \tag{4.17} \]

The conditions (4.15) and (4.17) are obtained respectively from (3.11) and (4.9) after replacing there the coordinates of the vector \(z\) by the coordinates of the vector \(z'\) and removing the conditions dependent on \(Z'\).

The place of the center conditions (4.15) and (4.16) in the system of all conditions for the existence of a center is clear from the following theorem, which is obtained by expressing the center conditions given in [24, 25] in terms of the coordinates of the vector \(z'\).

Theorem 4.4. For the existence at the origin of coordinates of a center of equation (4.7) for \(A'=\{2\}\), it is necessary and sufficient that at least one of the following three series of conditions be fulfilled:

1) \(z_{11}=0\);

2) \(\operatorname{Im}(z_{11}z'_{02})=\operatorname{Im}(z_{20}z_{02}^{\prime 3})=0\);

3) \(5\overline{z}_{11}+z'_{02}=5|z_{20}|-|z'_{02}|=0\),

and for \(A'=\{3\}\), at least one of the following two series of conditions:

1) \(\operatorname{Im}z_{12}=\operatorname{Im}(z_{21}z'_{03})=\operatorname{Im}(z_{30}z_{03}^{\prime 2})=0\);

2) \(z_{12}=10\overline{z}_{21}+z'_{03}=2|z_{21}|-|z_{30}|=0\).

The center conditions (4.17) have not previously occurred in the literature.

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Received by the editors
18 July 1965

Institute of Mathematics with Computing Center
Academy of Sciences of the Moldavian SSR