Malmquist-type theorems for cubic Hamiltonians

Abstract

The aim of this paper is to classify the cubic polynomials H(z,x,y)=∑j+k≤3ajk(z)xjyk

over the field of algebraic functions such that the corresponding Hamiltonian system x′=Hy, y′=−Hx has at least one transcendental algebroid solution. Ignoring trivial subcases, the investigations essentially lead to several non-trivial Hamiltonians which are closely related to Painlevé’s equations PI, PII, P34, and PIV

. Up to normalisation of the leading coefficients, common Hamiltonians are HI:HII/34:HIV:−2y3+12x2−zyx2y−12y2+12zy+κxx2y+xy2+2zxy+2κx+2λy13(x3+y3)+zxy+κx+λy,

but the zoo of non-equivalent Hamiltonians turns out to be much larger.