Instability of isolated equilibria of dynamical systems with invariant measure in spaces of odd dimension

We discuss the conjecture asserting that isolated equilibrium states of autonomous systems admitting invariant measures are unstable in spaces of odd dimension. This conjecture is proved for systems for which quasihomogeneous truncations with isolated singularities can be found. We consider a counterexample in the class of systems with infinitely differentiable right-hand sides and zero Maclaurin series at the equilibrium state. Translated fromMatematicheskie Zametki, Vol. 65, No. 5, pp. 674–680, May, 1999.