On the Birkhoff transformation in the case of complete degeneracy of the quadratic part of the Hamiltonian

A time-periodic system with one degree of freedom is investigated. The system is assumed to have an equilibrium position, in the vicinity of which the Hamiltonian is represented as a convergent series.This series does not contain members of the second degree, whereas the members to some finite degree $\ell$ do not depend explicitly on time. The algorithm for constructing a canonical transformation is proposed that simplifies the structure of the Hamiltonian in members to degree $\ell$, inclusive. As an application, a special case is considered when the expansion of the Hamiltonian begins with members of the third degree. For this case, sufficient conditions for instability of the equilibrium are obtained depending on the forms of the fourth and fifth degrees.