  A program is outlined concerning the set of all solutions of the hyperbolic
Ernst equation on a two-dimensional manifold whose underlying topological space
is the same as the domain of all Ernst potentials for colliding plane
gravitational wave pairs. The aim of the program is to construct and apply a
non-trivial extension of the group of Kinnersley-Chitre transformations. This
is to be done by employing the formalism of a homogeneous Hilbert problem. In
this first paper of a series, the aforementioned program is completely carried
out for the collinear polarization case.
