  We consider the Cauchy problem for the repulsive Vlasov-Poisson system in the
three dimensional space, where the initial datum is the sum of a diffuse
density, assumed to be bounded and integrable, and a point charge. Under some
decay assumptions for the diffuse density close to the point charge, under
bounds on the total energy, and assuming that the initial total diffuse charge
is strictly less than one, we prove existence of global Lagrangian solutions.
Our result extends the Eulerian theory of [16], proving that solutions are
transported by the flow trajectories. The proof is based on the ODE theory
developed in [8] in the setting of vector fields with anisotropic regularity,
where some components of the gradient of the vector field is a singular
integral of a measure.
