We consider four dimensional U(N) N = 4 SYM theory interacting with a 3d N = 4 theory living on a co dimension-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 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
