  Exceptional orthogonal polynomials are families of orthogonal polynomials
that arise as solutions of Sturm-Liouville eigenvalue problems. They generalize
the classical families of Hermite, Laguerre, and Jacobi polynomials by allowing
for polynomial sequences that miss a finite number of "exceptional" degrees. In
this paper we introduce a new construction of multi-parameter exceptional
Legendre polynomials by considering the isospectral deformation of the
classical Legendre operator. Using confluent Darboux transformations and a
technique from inverse scattering theory, we obtain a fully explicit
description of the operators and polynomials in question. The main novelty of
the paper is the novel construction that allows for exceptional polynomial
families with an arbitrary number of real parameters.
