Generalized spinning particles on $${\mathcal {S}}^2$$ in accord with the Bianchi classification

Motivated by recent studies of superconformal mechanics extended spin degrees freedom, we construct minimally superintegrable models spinning particles on 2-sphere, the freedom which are represented a 3-vector obeying structure relations 3d real Lie algebra. Generalisations involving an external field Dirac monopole, or motion group manifold SU(2), scalar potential giving rise to two quadratic constants discussed. A procedure how build similar extensions, rely upon d=4,5,6 algebras, is elucidated.