<p>We investigate the problem of small motions and normal oscillations of a viscous fluid. The free surface contains heavy particles of some substance. These particles do not interact with each other during the free surface oscillations, or their interaction is negligible. We reduce the original initial-boundary value problem to the Cauchy problem for a first-order differential equation in a Hilbert space. After a detailed study of the properties of the operator coefficients, we prove a theorem on the solvability of the resulting Cauchy problem. Based on this, we find sufficient conditions for the existence of a solution to the initial-boundary value problem describing the evolution of the original hydraulic system. We prove statements regarding the structure of the problem spectrum and the basis property of the system of eigenfunctions.</p>