  de la Pe\~na 1980 and Puthoff 1987 show that circular orbits in the hydrogen problem of Stochastic Electrodynamics are stable. Though the Cole-Zou 2003 simulations support the stability, our recent numerics always lead to self-ionisation. Here the de la Pe\~na-Puthoff argument is extended to elliptic orbits. For very eccentric orbits with energy close to zero and angular momentum below some not-small value, there is on the average a net gain in energy for each revolution, which explains the self-ionisation. Next, an $1/r^2$ potential is added, which could stem from a dipolar deformation of the nuclear charge by the electron at its moving position. This shape retains the analytical solvability. When it is enough repulsive, the ground state of this modified hydrogen problem is predicted to be stable. The same conclusions hold for positronium. 
On the stability of classical orbits of the hydrogen ground state in   Stochastic Electrodynamics