In this paper, we consider the natural complex Hamiltonian systems with homogeneous potential $V\left(q\right)$, $q\in {ℂ}^n$, of degree $k\in {\mathbb{Z}}^{☆}$. The known results of Morales and Ramis give necessary conditions for the complete integrability of such systems. These conditions are expressed in terms of the eigenvalues of the Hessian matrix $V^{\prime \prime }\left(c\right)$ calculated at a non-zero point $c\in {ℂ}^n$, such that $V^{\prime }\left(c\right)=c$. The main aim of this paper is to show that there are other obstructions for the integrability which appear if the matrix $V^{\prime \prime }\left(c\right)$ is not diagonalizable....