Exceptional orthogonal polynomials, exactly solvable potentials and supersymmetry

We construct two new exactly solvable potentials giving rise to bound-state solutions to the Schrödinger equation, which can be written in terms of the recently introduced Laguerre- or Jacobi-type X₁ exceptional orthogonal polynomials. These potentials, extending either the radial oscillator or the Scarf I potential by the addition of some rational terms, turn out to be translationally shape invariant as their standard counterparts and isospectral to them.