Application of the Kovacic algorithm to the study of the motion of a heavy rigid body with a fixed point in the Hess case

In 1890, W. Hess found a new particular case of the integrable Euler–Poisson equations of the motion of a heavy rigid body with a fixed point. In 1892, P. A. Nekrasov proved that the solution of the problem of the motion of a heavy rigid body with a fixed point under the Hess conditions can be reduced to integrating a second-order linear equation with variable coefficients. In this paper, we derive the corresponding second-order equation and reduce its coefficients to the rational form. Then, using the Kovacic algorithm, we examine the existence of Liouville solutions of the corresponding second-order linear equation. We prove that Liouville solutions can exist only in two cases: in the case corresponding to the Lagrange case of the motion of a rigid body with a fixed point and in the case where the area integral is equal to zero.