Consider a differential equation with partial derivatives ∂tu(t; x) = {A0(t; i∂x) + A1(t, x; i∂x)}u(t; x), (t; x) ∈ Π(0;T], (1) where u is an unknown function, ΠQ = {(t; x) : t ∈ Q, x ∈ Rn}, and A0(t; i∂x) =  |k|≤p a0,k(t)i|k|∂k x, A1 (t, x; i∂x) =  |k|≤p1 a1,k(t; x)i|k|∂k x are differential expressions of orders p and p1, respectively. We assume that Eq. ∂tu(t; x) = A0(t; i∂x)u(t; x), (t; x) ∈ Π(0;T], (2) is parabolic according to Shilov on the set Π[0;T] with the parabolicity index h, 0 < h ≤ p, and genus μ < 0, and the order p1 of the group of junior members of the Eq. (1) is less than h [1]: 0 ≤ p1 < h. Using the properties of the fundamental solution of the Cauchy problem for Eq. (3) studied in [2], the correctness of the following statement is proved. Theorem. Let the coefficients a0,k(t) and a1,k(t; x) of the Eq. (1) on the set Π[0;T] be continuous with respect to the variable t, are infinitely differentiable with respect to the variable x and are bounded together with their derivatives. Then, for Eq. (1), there exists a fundamental solution Z(t, x; τ, ξ) of the Cauchy problem, which is differentiable with respect to the variable t and infinitely differentiable with respect to each of the variables x and ξ. The following assessments are also correct: ∃δ > 0 ∀{r, q} ⊂ Zn + ∃c > 0 ∀ 0 ≤ τ < t ≤ T ∀{x; ξ} ⊂ Rn : |∂r ξ ∂qx Z(t, x; τ, ξ)| ≤ c(t − τ )−n+|r+q| h e−δ |x−ξ| λ (t−τ)γ (here |x| λ := |x1|λ + . . . + |xn|λ, λ := 1 1−μ/h and γ := 1 h−μ ). This information about the Z function is important for building the classical theory of the Cauchy problem for parabolic equations with negative genus and variable coefficients.