Abstract <p>In this paper, one of generalized Vakhnenko–Parkes’ family equations is considered describing the propagation of short-wave disturbances in relaxing media, taking into account the dependence of the wave velocity on the amplitude. A general quadrature solution is obtained for the equation under consideration by reducing it to an ordinary differential equation using the traveling wave variables. Some formal exact solutions of the initial equation are found. The periodic exact solutions are expressed in terms of Jacobi elliptic functions. An explicit solution is also presented, expressed in the terms of a power function of spatial and temporal variables. The obtained exact solutions can be used as the test functions when analyzing the results of a numerical simulation of the processes in the relaxing media described by Vakhnenko–Parkes type equations.</p>