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.