The paper studies the integrability of Hamiltonian systems \[ H(q,p) = \frac{1}{2} \sum_{i=1}^n p_i^2 + V(q), \quad q,p \in \mathbb C^n \] where \(V(q)\) is a homogeneous function of degree \(k \in \mathbb Z \setminus \{0\}\) with \(|k| \geq 3\). In particular, it investigates second-order variational equations along a particular solution given by \(q_0(t) = \varphi(t)d\), \(\tfrac12 \dot\varphi(t) +k\varphi(t)^k=e\), where \(d \in \mathbb C^n \setminus \{0\}\) is a solution of \(V'(d)=d\), and \(e\) is a constant of energy. The paper’s major results give strong necessary conditions guaranteeing that the Galois groups of specific associated systems are virtually abelian and thus necessary conditions for integrability. The formulated conditions give effective algorithms to test such possibilities in particular cases.