  For $x\in End(K^n)$ satisfying $x^2 = 0$ let $F_x$ be the variety of full
flags stable under the action of $x$ (Springer fiber over $x$). The full
classification of the components of $F_x$ according to their smoothness was
provided in a paper of Fresse-Melnikov in terms of both Young tableaux and link
patterns. Moreover in a paper of Fresse the purely combinatorial algorithm to
compute the singular locus of a singular components of $F_x$ is provided.
However this algorithm involves the computation of the graph of the component,
and the complexity of computations grows very quickly, so that in practice it
is impossible to use it. In this paper, we construct another algorithm, derived
from the algorithm of Fresse, providing all the components of the singular
locus of a singular component of $F_x$ in terms of link patterns constructed
straightforwardly from its link pattern.
