Geometric transformations and new integrable problems of rigid body dynamics

The problem of motion of a rigid body about a fixed point under the action of conservative forces is considered in the case admitting a linear integral but no axis of symmetry-neither in space nor in the body-is present. A simple transformation of the configuration space is used to reduce the problem of motion of the body to another problem concerning the same body under a system of axisymmetric forces. This analogy enables the construction of several new integrable cases of the first problem by transforming certain known ones of the second. The new cases usually involve singular potential terms. Integrals of motion and physical interpretation are given explicitly for one generally integrable case. Other general and conditional cases are pointed out.