Dynamics on a submanifold: intermediate formalism versus Hamiltonian reduction of Dirac bracket, and integrability

We consider Hamiltonian formulation of a dynamical system forced to move on a submanifold $G_\alpha(q^A)=0$. If for some reasons we are interested in knowing the dynamics of all original variables $q^A(t)$, the most economical would be a Hamiltonian formulation on the intermediate phase-space submanifold spanned by reducible variables $q^A$ and an irreducible set of momenta $p_i$, $[i]=[A]-[\alpha]$. We describe and compare two different possibilities for establishing the Poisson structure and Hamiltonian dynamics on an intermediate submanifold: Hamiltonian reduction of the Dirac bracket and intermediate formalism. As an example of the application of intermediate formalism, we deduce on this basis the Euler-Poisson equations of a spinning body, establish the underlying Poisson structure, and write their general solution in terms of the exponential of the Hamiltonian vector field.