Solvable rational extensions of the isotonic oscillator

Highlights: → We obtain in a new way the solvable rational extensions of the isotonic oscillator. → The method is systematic without resorting to any ansatz. → We use a generalization of the SUSY quantum partnership to excited states. → They are regularized by specific discrete symmetries of the potential. → The proof of the shape invariance of the extensions is direct. - Abstract: Combining recent results on rational solutions of the Riccati-Schroedinger equations for shape invariant potentials to the finite difference Baecklund algorithm and specific symmetries of the isotonic potential, we show that it is possible to generate the three infinite sets (L1, L2 and L3 families) of regular rational solvable extensions of this potential in a very direct and transparent way.