Two linear equations in formal power series

The equations gradφ(x)]^TF(x)=h(x) and F(ψ(x))–ψ(x) are considered. They arise in the stability theory of differential and difference equations. The scalar function h(x) is a given, and the function ψ(x) an unknown, formal power series in the n indeterminates x=(x₁,…,x_n)^T, and h(0)=ψ=0; the elements of the n×n matrix F(x) are also formal power series in x, F(0)=0. It is shown that the solvability of both equations depends on the eigenvalues of the Jacobian F_x(0).