On the existence of new integrable cases for Euler-Poisson equations in Newtonian fields

In this article, the rotational motion of a non-symmetric rigid body about a fixed point under the action of multi-Newtonian centers located on one fixed axis is considered. The aim of the work is to derive a technique to obtain such new (particular) integrable cases for the rotational motion of a rigid body in newtonian field of forces whose centers located on the vertical fixed axis. This technique reduces the governing equations and their known first integrals describing the motion to the unknown fourth integral and new system of differential equations. The well-known fourth integral cases are tested to verify the validity of the derived system and some new (particular) integrable cases are obtained with their geometrical interpretations of motion. The particular scope from this work is to extend Arkangel’skii’s theorem and the other related theorems to obtain such new integrable cases of single valuedness of solutions for this problem and its related special cases.