Malmquist-Type Theorems for Cubic Hamiltonians

Abstract The aim of this paper is to classify the cubic polynomials $$\begin{aligned} H(z,x,y)=\sum _{j+k\le 3}a_{jk}(z)x^jy^k \end{aligned}$$ H ( z , x y ) = ∑ j + k ≤ 3 a jk over field algebraic functions such that corresponding Hamiltonian system $$x'=H_y,$$ ′ $$y'=-H_x$$ - has at least one transcendental algebroid solution. Ignoring trivial subcases, investigations essentially lead several non-trivial Hamiltonians which are closely related Painlevé’s equations $$\mathrm{P_{I}}$$ P I , $$\mathrm{P_{II}}$$ II $$\mathrm{P_{34}}$$ 34 and $$\mathrm{P_{IV}}$$ IV . Up normalisation leading coefficients, common \begin{array}{rl} \mathrm{H_I}:&{}-2y^3+\frac{1}{2}x^2-zy\\ \mathrm{H_{II/34}}:&{} x^2y-\frac{1}{2}y^2+\frac{1}{2}zy+\kappa x\\ \mathrm{H_{IV}}:&{}\begin{array}{l} x^2y+xy^2+2zxy+2\kappa x+2\lambda y\\ \frac{1}{3}(x^3+y^3)+zxy+\kappa x+\lambda y,\end{array} \end{array} : 2 1 / κ λ but zoo non-equivalent turns out be much larger.