Integrable corners in the space of Gukov-Witten surface defects

We investigate integrability properties of Gukov-Witten 1/2-BPS surface defects in $SU\left(N\right)$ $N=4$ super-Yang-Mills (SYM) theory in the large-N limit. We demonstrate that ordinary Gukov-Witten defects, which depend on a set of continuous parameters, are not integrable except for special sub-sectors. In contrast to these, we show that rigid Gukov-Witten defects, which depend on a discrete parameter but not on continuous ones, appear integrable in a corner of the discrete parameter space. Whenever we find an integrable sector, we derive a closed-form factorised expression for the leading-order one-point function of unprotected operators built out of the adjoint scalars of $N=4$ SYM theory. Our results raise the possibility of finding an all-loop formula for one-point functions of unprotected operators in the presence of a rigid Gukov-Witten defect at the corner in parameter space.