  We consider four dimensional $U(N)$ $\mathcal N=4$ SYM theory interacting
with a 3d $\mathcal N=4$ theory living on a codimension-one interface and
holographically dual to the D3-D5 system without flux. Localization captures
several observables in this dCFT, including its free energy, related to the
defect expectation value, and single trace $\frac{1}{2}$-BPS composite scalars.
These quantities may be computed in a hermitian one-matrix model with
non-polynomial single-trace potential. We exploit the integrable Volterra
hierarchy underlying the matrix model and systematically study its $1/N$
expansion at any value of the 't Hooft coupling. In particular, the strong
coupling regime is determined -- up to non-perturbative exponentially
suppressed corrections -- by differential relations that constrain higher order
terms in the $1/N$ expansion. The analysis is extended to the model with
$SU(N)$ gauge symmetry by resorting to the more general Toda lattice equations.
