  We describe a generalization of the notion of a Hilbert space model of a
function in the Schur class of the bidisc. This generalization is well adapted
to the investigation of boundary behavior at a mild singularity of the function
on the 2-torus. We prove the existence of a generalized model with certain
properties corresponding to such a singularity and use this result to solve two
function-theoretic problems. The first of these is to characterise the
directional derivatives of a function in the Schur class at a singular point on
the torus for which the Carath\'eodory condition holds. The second is to obtain
a representation theorem for functions in the two-variable Pick class analogous
to the refined Nevanlinna representation of functions in the one-variable Pick
class.
