The authors investigate a second order Hamiltonian system of the form \(\ddot u + \nabla V(u) = 0\) in \(\mathbb{R}^n\), where the potential has a unique strict global maximum at the origin \(p\) and a singular set \(S \not\ni p\) such that \(\mathbb{R}^n \backslash S\) is open, path-connected and has non-trivial fundamental group \(\pi_1 = G\). Some additional conditions on behaviour of \(V\) near \(S\) and \(p\) are also assumed so that the class of systems under consideration includes, for instance, the classical \(n\)-body problem. Under some suitable conditions on \(V\) and \(G\) the authors prove the existence of multiple homoclinics, each one belonging to a different class of \(G\).