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In this paper non-isolated equilibria of conservative mechanical systems are studied. Under mild non-degeneracy conditions such equilibria are shown to be unstable in the sense of Liapunov.

In this work the instability of a degenerate equilibrium position is studied through the formal series solutions. The inversion of the Lagrange-Dirichlet stability theorem is proved in the case of two zero eigenvalues and a nondegenerate Newton′s diagram. This case includes all singularities appearing in a nonremovable way in families depending on not more than 16 parameters.