Let G(t, x) be the Green's function of a parabolic differential operator ∂/∂t + P(- i∂/∂x). In a previous article of the authors (Math. USSR Sb. 20 (1973), 519-542) estimates for G are obtained by means of a convex function νp invariantly defined by P, and the saddle points are distinguished under the assumption that νp is smooth. In the present paper the question of the existence of a finite number of saddle points is studied without assuming the smoothness of νp; an example of a polynomial P is constructed for which the function νp is not smooth. It is shown that for almost all polynomials P the function νp is strictly convex almost everywhere. Bibliography: 13 items.