On the stability criteria in A. M. Lyapunov's article "Analysis of one of the partucular cases of the problem of stability of motion"

We consider the problem of stability of an equilibrium of an autonomous system of differential equations in the critical case of a double zero root (Jordan cell). Using his first method A. M. Lyapunov [1] found a criterion of stability for any nonlinear degeneration of the system. He addressed most of the subcases of the problem, using his direct method as well. The Lyapunov functions for all remaining subcases were not constructed yet. In this paper we found these functions in a few cases. This enables us to to propose a new algorithm of determination of equilibrium stability for some degenerations of system. This algorithm is specified by some algebraic operations over the Taylor series coefficients of the system whereas the algorithm by A. M. Lyapunov demands the calculation of quadratures.