with constant coefficients. We have obtained the solution of this system using Fourier transforms for the space coordinates and Laplace transform for time. From theorems about application of Fourier and Laplace transforms for system of linear partial differential equations with constant coefficients we see that in case if initial velocity and applied force are smooth enough functions decreasing in infinity, then the solution of such system is also a smooth function. (Corresponding theorems are presented in S. Bochner $\cite{SB59}$, V.P. Palamodov $\cite{VP70}$, G.E. Shilov $\cite{gS01}$, L. Hormander $\cite{LH83}$, S. Mizohata $\cite{SM73}$, J.F. Treves $\cite{JFT61}$). Result of this stage is the integral equation for vector-function of velocity.

**Second stage** (sections 4, 5) - we have introduced perfect spaces of functions and vector-functions (I.M. Gel’fand, G.E. Chilov $\cite{GC68}$), in which we have looked for the solution of the problem. We have demonstrated equivalence of the solution of the Cauchy problem in forms of differential and integral equations.

**Third stage** (section 6) - we have divided all parts of integral equation by an appropriate constant , and received equivalent integral equation. We have also correspondingly replaced integration variables in integral operators. This newly received equivalent integral equation allowed us to analyze the Cauchy 