Sufficient conditions for the existence of a center in a second-order nonlinear dynamical system in a critical case

We study an autonomous nonlinear system of second-order differential equations whose linear approximation matrix has a pair of purely imaginary eigenvalues and whose nonlinear part can be represented as the sum of forms of order ${\geqslant}2$ with respect to the components of the phase vector. We obtain sufficient conditions for the existence of a center or focus in a neighborhood of the zero solution.