  We consider coefficient bodies $\mathcal M_n$ for univalent functions. Based
on the L\"owner-Kufarev parametric representation we get a partially integrable
Hamiltonian system in which the first integrals are Kirillov's operators for a
representation of the Virasoro algebra. Then $\mathcal M_n$ are defined as
sub-Riemannian manifolds. Given a Lie-Poisson bracket they form a grading of
subspaces with the first subspace as a bracket-generating distribution of
complex dimension two. With this sub-Riemannian structure we construct a new
Hamiltonian system and calculate regular geodesics which turn to be horizontal.
Lagrangian formulation is also given in the particular case $\mathcal M_3$.
