Systems of $sl(2,\mathbb{C})$ tops as two-particle systems

We show that Euler–Arnold tops on the algebra $sl(2,\mathbb C)$ are equivalent to a two-particle system of Calogero type. We show that an arbitrary quadratic Hamiltonian of an $sl(2,\mathbb C)$ top can be reduced to one of the three canonical Hamiltonians using the automorphism group of the algebra. For each canonical Hamiltonian, we obtain the corresponding two-particle system and write the bosonization formulas for the coadjoint orbits explicitly. We discuss the relation of the obtained formulas to nondynamical Antonov–Zabrodin–Hasegawa $R$-matrices for Calogero–Sutherland systems.