Integrability conditions of geodesic flow on homogeneous Monge manifolds

We prove a meromorphic integrability criterion for the geodesic flow of an algebraic manifold of the form zp−f(x1,…,xn)=0 with the induced metric of Cn+1 and f a homogeneous rational function, using a parallel between the properties of such algebraic manifolds and homogeneous potentials. We then apply this criterion to the manifolds of the form z=λ1xk1+⋯+λnxkn, k∈Z+, and xnymzl=1,n,m,l∈Z, and prove that their geodesic flow is not integrable except for some given exceptional cases.