Analysis of stability and stabilization of nonlinear systems via decomposition

We establish necessary and sufficient conditions for the solvability of a Lyapunov-type system of PDEs in the class of homogeneous functions. Using these, we propose an approach to studying the stability of an equilibrium of an essentially nonlinear system of ODEs in the critical case of $n$ zero roots and $n$ pure imaginary roots. The approach bases on decomposition of the system in question into two separate subsystems of half dimension.