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We study four problems in the dynamics of a body moving about a fixed point,providing a non-complex, analytical solution for all of them. For the firsttwo, we will work on the motion first integrals. For the symmetrical heavybody, that is the Lagrange-Poisson case, we compute the second and third Eulerangles in explicit and real forms by means of multiple hypergeometric functions(Lauricella, functions). Releasing the weight load but adding the complicationof the asymmetry, by means of elliptic integrals of third kind, we provide theprecession angle completing some previous treatments of the Euler-Poinsot case.Integrating then the relevant differential equation, we reach the finite polarequation of a special trajectory named the {\it herpolhode}. In the lastproblem we keep the symmetry of the first problem, but without the weight, andtake into account a viscous dissipation. The approach of first integrals is nolonger practicable in this situation and the Euler equations are faced directlyleading to dumped goniometric functions obtained as particular occurrences ofBessel functions of order $-1/2$.

Motions about a fixed point by hypergeometric functions: new non-complex analytical solutions and integration of the herpolhode