  We find, and analyse, the exact solution of two friendly directed walks,
modelling polymers, which interact with a wall via contact interactions. We
specifically consider two walks that begin and end together so as to imitate a
polygon. We examine a general model in which a separate interaction parameter
is assigned to configurations where both polymers touch the wall
simultaneously, and investigate the effect this parameter has on the
integrability of the problem. We find an exact solution of the generating
function of the model, and provide a full analysis of the phase diagram that
admits three phases with one first-order and two second-order transition lines
between these phases. We argue that one physically realisable model would see
two phase transitions as the temperature is lowered.
