On conserved quantities of certain cohomogeneity one Ricci-flat equations

We give a classification of continuously differentiable on-shell integrals which are linear or quadratic in momenta for a Hamiltonian system with constraint associated to the Ricci-flat cohomogeneity one Einstein equations for two special classes of principal orbits/hypersurfaces. Relations to known on-shell integrals and to Painlev analysis are discussed.