  We study classical integrable systems based on the Alekseev-Meinrenken
dynamical r-matrices corresponding to automorphisms of self-dual Lie algebras,
${\cal G}$. We prove that these r-matrices are uniquely characterized by a
non-degeneracy property and apply a construction due to Li and Xu to associate
spin Calogero type models with them. The equation of motion of any model of
this type is found to be a projection of the natural geodesic equation on a Lie
group $G$ with Lie algebra ${\cal G}$, and its phase space is interpreted as a
Hamiltonian reduction of an open submanifold of the cotangent bundle $T^*G$,
using the symmetry arising from the adjoint action of $G$ twisted by the
underlying automorphism. This shows the integrability of the resulting systems
and gives an algorithm to solve them. As illustrative examples we present new
models built on the involutive diagram automorphisms of the real split and
compact simple Lie algebras, and also explain that many further examples fit in
the dynamical r-matrix framework.
