  Starting from a system of $N$ radial Schr\"odinger equations with a vanishing
potential and finite threshold differences between the channels, a coupled $N
\times N$ exactly-solvable potential model is obtained with the help of a
single non-conservative supersymmetric transformation. The obtained potential
matrix, which subsumes a result obtained in the literature, has a compact
analytical form, as well as its Jost matrix. It depends on $N (N+1)/2$
unconstrained parameters and on one upper-bounded parameter, the factorization
energy. A detailed study of the model is done for the $2\times 2$ case: a
geometrical analysis of the zeros of the Jost-matrix determinant shows that the
model has 0, 1 or 2 bound states, and 0 or 1 resonance; the potential
parameters are explicitly expressed in terms of its bound-state energies, of
its resonance energy and width, or of the open-channel scattering length, which
solves schematic inverse problems. As a first physical application,
exactly-solvable $2\times 2$ atom-atom interaction potentials are constructed,
for cases where a magnetic Feshbach resonance interplays with a bound or
virtual state close to threshold, which results in a large background
scattering length.
