  We study representations of inner functions on the bidisc from a fractional
linear transformation point of view, and provide sufficient conditions, in
terms of colligation matrices, for the existence of two-variable inner
functions. Here the sufficient conditions are not necessary in general, and we
prove a weak converse for rational inner functions that admit one variable
factorization.
  We present a complete classification of de Branges-Rovnyak kernels on the
bidisc (which equally works in the setting of polydisc and the open unit ball
of $\mathbb{C}^n$, $n \geq 1$). We also classify, in terms of Agler kernels,
two-variable Schur functions that admit one variable factor.
