  We construct a new, two-parametric family of integrable models and reveal
their underlying duality symmetry. A modular subgroup of this duality is shown
to connect non-interacting modes of different systems. We apply the new
solution and duality to a Richardson-Gaudin model and generate a novel
integrable system termed the $s$-$d$ wave Richardson-Gaudin-Kitaev interacting
chain, interpolating $s$- and $d$- wave superconductivity. The phase diagram of
this model has a topological phase transition that can be connected to the
duality, where the occupancy of the non-interacting mode serves as a
topological order parameter.
