On conditions for algebraic integrability of systems with a homogeneous potential (Q1406377)

The main result of the paper is as follows: the Hamiltonian system with Hamiltonian \[ H = \frac12\big(y_1^2 + y_2^2\big) + ax_1^3 + bx_1^2x_2 + dx_2^3, \] where \(a\), \(b\), \(d\in \mathbb R\), is algebraically integrable only in the following cases: (1) \(b=0\); (2) \(a=0\), \(\,2b=d\); (3) \(b=3d\).