  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 proves 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 states and was already known. The second emerges on
the first excited states, 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.
