Brownian motion in the quadrant with oblique repulsion from the sides

We consider the problem of strong existence and uniqueness of a Brownian motion forced to stay in the quadrant by an electrostatic repulsion from the sides that works obliquely. The results are reminiscent of the study of a Brownian motion with oblique reflection in a wedge. Actually, the same skew symmetry condition is involved when looking for a stationary distribution in product form. the terms of the product are now gamma distributions in place of exponential ones. An associate purely deterministic problem is also considered.