Hydrodynamic-type systems describing 2-dimensional polynomially   integrable geodesic flows
  Starting from a homogeneous polynomial in momenta of arbitrary order we extract multi-component hydrodynamic-type systems which describe 2-dimensional geodesic flows admitting the initial polynomial as integral. All these hydrodynamic-type systems are semi-Hamiltonian, thus implying that they are integrable according to the generalized hodograph method. Moreover, they are integrable in a constructive sense as polynomial first integrals allow to construct generating equations of conservation laws. According to the multiplicity of the roots of the polynomial integral, we separate integrable particular cases. 
Gianni Manno, Maxim V. Pavlov
math.DG