Special classes of solutions to the equations of horizontal-displacement fluid motion

We construct a special class of solutions to nonlinear integrodifferential equations describing horizontal-displacement motion of ideal fluid in an open channel in the shallow water approximation. This class of solutions is characterized by a linear relation among Riemann integral invariants and is determined by a hyperbolic system of two differential equations with two parameters. In the class of traveling waves we construct exact solutions, including arbitrary functions, to the equations of motion continuously adjacent to a given displacement flow.