Examples of integrable equations of motion of a five-dimensional rigid body in the presence of internal and external force fields

In the study of integrable systems that describe multidimensional rigid bodies in nonconservative force fields, two approaches are used. The first approach is concerned with systems in which the nonconservativity of force fields is related to additional coefficients in the cinematical relations; note that $n=5$ and $n=6$ are special cases. The second approach is based on the simultaneous influence of two force fields: internal (conservative) and external (nonconservative). This paper is devoted to the special case where $n=5$.