  Given a singular surface X, one can extract information on it by
investigating the fundamental group $\pi_1(X - Sing_X)$. However, calculation
of this group is non-trivial, but it can be simplified if a certain invariant
of the branch curve of X - called the braid monodromy factorization - is known.
This paper shows, taking the Cayley cubic as an example, how this fundamental
group can be computed by using braid monodromy techniques and their liftings.
This is one of the first examples that uses these techniques to calculate this
sort of fundamental group.
