First integrals of the nonlinear parabolic equation \begin{equation}
\frac{\partial u(t,x)}{\partial t}=\mathfrak U(u),\qquad x\in R^n,\quad t>t_0,
\end{equation}are considered, i.e. functionals$G(t,u)$that are constant on solutions$u(t,x)$of (1):$G(t,u(t,x))=\mathrm{const}$. Every first integral satisfies a first-order variational differential equation. A solution of the Cauchy problem is constructed for this equation. The method of constructing these solutions, i.e. first integrals, affords a number of corollaries concerning statistical characteristics of solutions of (1).Bibliography: 4 titles.