Analytical solutions for Euler parameters

Two new analytical solutions for Poinsot motion in terms of Euler parameters are derived. The first solution is a straightforward ‘universal’ (no branches) time series practical for short time motion calculations or as a basis for analytical continuation. The second, more involved solution is also universal but is not restricted to short times; it is in terms of circular, hyperbolic, and elliptic functions and elliptic integrals.