Let $\Gamma$ be the unit circle and let $L^k$ ( $k=1,2,\dots$ ) be the Hilbert space of vector functions $f(\zeta)=\{f_j(\zeta)\}_{j=1}^k$ with coordinates in $L_2(\Gamma)$ . Theorem. {\it Let$a(\zeta),b(\zeta)$$(\zeta\in\Gamma)$be$m\times n$matrices with elements continuous on$\Gamma$. In order for the singular integral operator$T,$from$L^n$to$L^m,$$$ (Tf)(\zeta)=c(\zeta)f(\zeta)+\frac{d(\zeta)}{\pi i}\int_\Gamma\frac{f(z)}{z-\zeta}\,dz\qquad(f\in L^n) $$to be normally solvable it is necessary and sufficient for the following two conditions to be satisfied}.a)The rank of each of the matrices$c(\zeta)+d(\zeta)$and$c(\zeta)-d(\zeta)$is independent of$\zeta$on the unit circumference.b) {\it$\inf_{x\in(\operatorname{Ker}\,T)^\perp,\,\|x\|=1}\{\rho(Px,\operatorname{Ker}aI)+\rho(Qx,\operatorname{Ker}bI)\}>0.$}By$P$we denote the orthogonal projector in$L^n$defined by $(Pf)(\zeta)=\frac12f(\zeta)+\frac1{2\pi i}\int_\Gamma\frac{f(z)}{z-\zeta}\,dz$ ($f\in L^n$),$Q=I-P$. The conditions a) and b) are independent.The theorem is applicable to equations of Wiener–Hopf type.Bibliography: 11 titles.