The Euler−Poinsot rigid body
motion is a standard mechanical system and it is a model for left-invariant Riemannian
metrics on SO(3). In this article using the
Serret−Andoyer variables we
parameterize the solutions and compute the Jacobi fields in relation with the conjugate
locus evaluation. Moreover, the metric can be restricted to a 2D-surface, and the
conjugate points of this metric are evaluated using recent works on surfaces of
revolution. Another related 2D-metric on S2 associated to the dynamics of spin particles with
Ising coupling is analysed using both geometric techniques and numerical simulations.