  This article is the continuation of our first work on the determination of
the cases where there is equality in Courant's Nodal Domain theorem in the case
of a Robin boundary condition (with Robin parameter $h$). For the square, our
first paper focused on the case where $h$ is large and extended results that
were obtained by Pleijel, B\'erard--Helffer, for the problem with a Dirichlet
boundary condition. There, we also obtained some general results about the
behaviour of the nodal structure (for planar domains) under a small deformation
of $h$, where $h$ is positive and not close to $0$. In this second paper, we
extend results that were obtained by Helffer--Persson-Sundqvist for the Neumann
problem to the case where $h>0$ is small.
