  We consider the Springer fiber ${\mathcal B}_u$ corresponding to a nilpotent
endomorphism $u$ of nilpotent order 2. As a first result, we give a description
of the elements of a given component of ${\mathcal B}_u$ which are fixed by the
action of the standard torus relative to some Jordan basis of $u$. By using
this result, we establish a necessary and sufficient condition of singularity
for the components of ${\mathcal B}_u$.
