Nonlinear differential operators of the first and the second orders, possessing invariant linear spaces of the maximal dimension

In connection with the approach to the construction of explicit solutions for nonlinear partial differential equations, proposed by S. S. Titov and V. A. Galaktionov, the problem of description of nonlinear differential operators $F[y(x)]$ possessing finite-dimensional invariant linear spaces arises. It was proved previously that for the $m$-th order operators the dimension of an invariant space cannot еxceed $2m+1$. In the present paper we consider the cases, when this value is attained. The first and the second order operators are studied. It is shown that they are quadratic in $y$. The full description of the first order operators and of the second order quadratic operators with constant coefficients is obtained.