The class of equations considered in the paper is a combination of two classes of equations: a
degenerate parabolic equation of the Kolmogorov type and a parabolic equation with increasing
coefficients in the group of younger members.Such a combination occurs in the problems of the
theory of stochastic processes where, in the case of a normal Markov process, the Kolmogorov-
Fokker-Planck equation has a similar form. The coefficients of this equations are constant in
the group of principal terms and ones are increasing functions in the group of lowest terms. The
article is devoted to the study of the properties of the volume potential, the kernel of which
is the fundamental solution of the Cauchy problem for such an equation. Estimates of the
fundamental solution of the Cauchy problem have a more complex structure than in the case
of the classical Kolmogorov equation. These properties concern the existence of the derivatives
included in the equation. They are used to establish theorems on the integral representations of
solutions of the Cauchy problem and theorems on the correct solvability of the Cauchy problem
in the corresponding classes of functional spaces. Such studies are carried out in this work. The
obtained results are new and published for the first time.

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