  We start from a given one dimensional rationally extended potential
associated with $X_m$ exceptional orthogonal polynomials and using the idea of
supersymmetry in quantum mechanics, we obtain one continuous parameter
($\lambda$) family of rationally extended strictly isospectral potentials whose
solutions are also associated with Xm exceptional orthogonal polynomials. We
illustrate this construction by considering three well known rationally
extended potentials, two with pure discrete spectrum (the extended radial
oscillator and the extended Scarf-I) and one with both the discrete and the
continuous spectrum (the extended generalized Poschl-Teller) and explicitly
construct the corresponding one continuous parameter family of rationally
extended strictly isospectral potentials. Further, in the special case of
$\lambda = 0$ and $-1$, we obtain two new exactly solvable rationally extended
potentials, namely the rationally extended Pursey and the rationally extended
Abhrahm-Moses potentials respectively. We illustrate the whole procedure by
discussing in detail the particular case of the $X_1$ rationally extended one
parameter family of potentials including the corresponding Pursey and the
Abraham Moses potentials.
