An initial boundary value problem of elastic vibrations of ice in a channel caused by an external load motion is considered. The mathematical model is based on a system of differential equations that describes the oscillations of the ice cover and motion of liquids in the channel. The ice cover is modeled by an equation of a thin elastic plate. The function of the ice plate deflection satisfies fixed conditions on walls of the channel. The liquid is inviscid and incompressible. The fluid flow potential satisfies the Laplace equation, conditions of impermeability on the walls and channel bottom, and linearized dynamic and kinematic conditions on the ice-liquid interface. One of the fundamental points of the problem is the existence and uniqueness of solutions for the taken coupled system of equations. The paper investigates the problems of the solvability for the coupled dynamic equations for the fluid and the elastic plate. Algorithm for solving the problem and proving the existence of classical solutions is presented in paragraph 1. The initial problem is reduced by applying the Fourier transformation to the problem of oscillation profile across the channel which is solved by the normal mode method. The result is a system of linear differential equations for normal decomposition coefficients of ice deflection in normal modes. The classical solution uniqueness of the considered initial boundary value problem is proved in paragraph 2.