New families of conservative systems on S possessing an integral of fourth degree in momenta

H = ρ0(r, B1, B2)(r dφ + dr)− ρ1(r)(B1 sin 2φ+B2 cos 2φ)− ρ2(r) cosφ (2) where B1, B2 are constants and ρ0, ρ1, ρ2 are some functions. It has been shown in [5] that this family reduced to the case of Kovalevskaya when B1 = B2 = 0. We say that a conservative system on S is smooth if the Hamiltonian is a sum of a smooth Riemannian metric on S (kinetic energy) and a smooth function U on S (potential energy or simply potential). In this sence above examples of Kovalevskaya and Goryachev are smooth conservative systems on S. In this paper we proposed new examples of smooth conservative systems on S possessing an integral of fourth degree in momenta. In our examples we use the solution ψ0 of the following initial value problem ψψ + 2ψ − 3ψ = 0, ψ(0) = 0, ψ(0) = 1, ψ(0) = 0. (3)