  In classical numerical relativity, marginally outer trapped surfaces (MOTSs)
are the main tool to locate and characterize black holes. For five decades it
has been known that during a binary merger, a new outer horizon forms around
the initial apparent horizons of the individual holes once they are
sufficiently close together. However the ultimate fate of those initial
horizons has remained a subject of speculation. Recent axisymmetric studies
have shed new light on this process and this pair of papers essentially
completes that line of research: we resolve the key features of the
post-swallowing axisymmetric evolution of the initial horizons. This first
paper introduces a new shooting-method for finding axisymmetric MOTSs along
with a reinterpretation of the stability operator as the analogue of the Jacobi
equation for families of MOTSs. Here, these tools are used to study exact
solutions and initial data. In the sequel paper they are applied to black hole
mergers.
