  In a general setting of a Hamiltonian system with two degrees of freedom and
assuming some properties for the undergoing potential, we study the dynamics
close and tending to a singularity of the system which in models of $N$-body
problems corresponds to total collision. We restrict to potentials that exhibit
two more singularities that can be regarded as two kind of partial collisions
when not all the bodies are involved. Regularizing the singularities, the total
collision transforms into a 2-dimensional invariant manifold. The goal of this
paper is to prove the existence of different types of ejection-collision
orbits, that is, orbits that start and end at total collision. Such orbits are
regarded as heteroclinic connections between two equilibrium points and are
mainly characterized by the partial collisions that the trajectories find on
their way. The proof of their existence is based on the transversality of
2-dimensional invariant manifolds and on the behavior of the dynamics on the
total collision manifold, both of them are thoroughly described.
