  We consider the two dimensional unsteady Prandtl system. For a special class
of outer Euler flows and solutions of the Prandtl system, the trace of the
tangential derivative of the tangential velocity along the transversal axis
solves a closed one dimensional equation. First, we give a precise description
of singular solutions for this reduced problem. A stable blow-up pattern is
found, in which the blow-up point is ejected to infinity in finite time, and
the solutions form a plateau with growing length. Second, in the case where,
for a general analytic solution, this trace of the derivative on the axis
follows the stable blow-up pattern, we show persistence of analyticity around
the axis up to the blow-up time, and establish a universal lower bound of
$(T-t)^{7/4}$ for its radius of analyticity.
