Euler-Calogero-Moser model from $\mathrm{SU}\left(2\right)\mathrm{}$ Yang-Mills theory

The relation between the Euler-Calogero-Moser model and $\mathrm{SU}\left(2\right)\mathrm{}$ Yang-Mills mechanics, originating from the four-dimensional $\mathrm{SU}\left(2\right)\mathrm{}$ Yang-Mills theory under the supposition of spatial homogeneity of the gauge fields, is discussed in the framework of Hamiltonian reduction. Two kinds of reduction of the degrees of freedom are considered: due to gauge invariance and due to discrete symmetry. In the former case, it is shown that after elimination of the gauge degrees of freedom from the $\mathrm{SU}\left(2\right)\mathrm{}$ Yang-Mills mechanics the resulting unconstrained system represents the ${\mathrm{ID}}_3$ Euler-Calogero-Moser model with a certain external fourth-order potential. In the latter, the ${\mathrm{IA}}_6$ Euler-Calogero-Moser model embedded in a special external potential is considered, whose projection onto the invariant submanifold through the discrete symmetry coincides again with the $\mathrm{SU}\left(2\right)\mathrm{}$ Yang-Mills mechanics. Based on this connection, the equations of motion of the unconstrained $\mathrm{SU}\left(2\right)\mathrm{}$ Yang-Mills mechanics in the limit of zero coupling constant are presented in the Lax form.