On reduction of multidimensional first order equations with multihomogeneous function of derivatives

We present an analysis of solutions to multidimensional first order equation with an arbitrary number of independent variables. It is assumed that the nonlinear part of the equation is a multihomogeneous function of derivatives. The reduction of an original equation is executed for the class of solutions depending on linear combinations of initial variables, each of which contains only a certain subset of variables. It is shown that the reduced equation has solutions in the form of some arbitrary functions and solutions in the form of some generalized polynomials. We also consider the cases of additional, multiplicational and combined separation of variables.