The objective of this article is to analyze the integrability proper-ties of extremal solutions of Pontryagin Maximum Principle in the time min-imal control of a linear spin system with Ising coupling in relation with con-jugate and cut loci computations. Restricting to the case of three spins, the problem is equivalent to analyze a family of almost-Riemannian metrics on the sphere S 2 , with Grushin equatorial singularity. The problem can be lifted into a SR-invariant problem on SO(3), this leads to a complete understanding of the geometry of the problem and to an explicit parametrization of the extremals using an appropriate chart as well as elliptic functions. This approach is com-pared with the direct analysis of the Liouville metrics on the sphere where the parametrization of the extremals is obtained by computing a Liouville nor-mal form. Finally, an algebraic approach is presented in the framework of the application of differential Galois theory to integrability.