On the integrability of the Hamiltonian systems with homogeneous polynomial potentials

We summarize the known results on the integrability of the complex Hamiltonian systems with two degrees of freedom defined by the Hamiltonian functions of the form$$\begin{array}{} \displaystyle H=\frac{1}{2}\sum_{i=1}^{2}p_i^2+V(q_1,q_2), \end{array} $$where V(q1,q2) are homogeneous polynomial potentials of degree k.