Consider the system $$ \begin{cases} \dot{x}=f(x,\alpha), \\ \dot{\alpha}=0, \end{cases} $$ with $(x,\alpha)^T\in\mathbb{R}^{n+1}$. The system has a homoclinic orbit at $\alpha=0$ and we assume there is a curve of saddles which has the stable and unstable invariant manifolds $\mathcal{W}^s$ and $\mathcal{W}^u$.
In Kuznetsov's book, p. 231, it is written that $\mathcal{W}^s$ and $\mathcal{W}^u$ intersect transversally along the homoclinic orbit if and only if the Melnikov integral is non-zero. In the book one of the directions is proved, namely, if the intersection is non-transerval, then the integral is zero. However, it says nothing about the remaining direction, i.e., if the intersection is transversal, then the integral is zero. And I do not see that this is trivial... Does anyone have an idea how to prove that?