A criterion for the algebraic integrability of Hamiltonian systems with a fourth-degree homogeneous potential (Q1851513)

The following result is established. The Hamiltonian system \[ \dot x =\partial H/\partial y,\quad\dot y= -\partial H /\partial x, \] \(x = (x_1,x_2)\), \(y = (y_1,y_2)\), with the Hamiltonian \[ H = \tfrac12(y_1^2+y_2^2)+ax_1^4+bx_1^3x_2 + cx_1^2x_2^2 +fx_1x_3^3 + dx_2^4,\quad a,\,b,\,c,\,d,\,f\in \mathbb{R} \] is algebraically integrable only in the cases: 1. \(b=0\), \(c=0\); 2. \(b=0\), \(c=6a\), \(d=a\); 3. \(b=0\), \(c=2a\), \(d=a\).