Integrable systems on tangent bundle of multi-dimensional sphere

The systems which have finite-dimensional spheres as the space of positions, are arising in many problems of multi-dimensional dynamics. Accordingly, tangent bundles of those spheres become phase spaces of such systems. In the article activity of inductive transition in the system on tangent bundle of low-dimensional sphere under increase of its dimension and absence of force field is analyzed. At that, nonconservative fields of forces are presented with the presence of which the systems possess the complete choice of first integrals expressing in terms of finite combination of elementary functions and are, in general, the transcendental functions of its variables.