We explicitly give all stationary solutions to the focusing cubic NLS on the line, in the
presence of a defect of the type Dirac’s delta or delta prime. The models prove
interesting for two features: first, they are exactly solvable and all quantities can be
expressed in terms of elementary functions. Second, the associated dynamics is far from
being trivial. In particular, the NLS with a delta prime potential shows two symmetry
breaking bifurcations: the first concerns the ground state and was already known. The
second emerges on the first excited state, and up to now had not been revealed. We
highlight such bifurcations by computing the nonlinear and the no-defect limits of the
stationary solutions.