A four-direction cyclic random motion with constant finite speed v in the plane R2 driven by a homogeneous Poisson process of rate λ > 0 is studied. A fourth-order hyperbolic equation with constant coefficients governing the transition law of the motion is obtained. A general solution of the Fourier transform of this equation is given. A special non-linear automodel substitution is found reducing the governing partial differential equation to the generalized fourth-order ordinary Bessel differential equation, and the fundamental system of its solutions is explicitly given.