  We reformulate the Generalized Proudman--Johnson (GPJ) equation with
parameter a in Lagrangian variables, where it takes the form of an
inhomogeneous Liouville equation. This allows us to provide an explicitformula
for the flow map, up to the solution of an ODE. Depending on the parameter a,
we prove new criteria for global existence or formation of a finite-time
singularity. In particular, we show that there exist smooth initial data which
become singular in finite time for a > 1. Also, we give a physical derivation
of the GPJ equation for general parameter values of a.
