On generalized N. Kovalevski equations in two problems of rigid body dynamics

In this paper we consider the reduction of Kirchhoff–Poisson equations related to the problem of rigid body motion under the action of potential and gyroscopic forces and also equations of the problem of rigid body motion taking into account the Barnett–London effect. For the above-mentioned problems, we obtain analogues of N. Kovalevski equations. In addition, for the above-mentioned problems we obtain two new particular solutions to the polynomial class of Steklov–Kovalevski–Goryachev reduced differential equations. The polynomial solution of the problem of gyrostat motion under the action of potential and gyroscopic forces is characterized by the following property: the squares of the second and the third vector component of angular velocity are quadratic polynomials of the first vector component that is an elliptic function of time. A polynomial solution of the equation of rigid body motion in a magnetic field (taking into account the Barnett–London effect) is characterized by the fact that the square of the second vector component of the angular velocity is the second-degree polynomial, while the square of the third component is the fourth-degree polynomial of the first vector component. The former is found as a result of an elliptic integral inversion.