  We analyze the fine structure of Clark measures and Clark isometries
associated with two-variable rational inner functions on the bidisk. In the
degree (n,1) case, we give a complete description of supports and weights for
both generic and exceptional Clark measures, characterize when the associated
embedding operators are unitary, and give a formula for those embedding
operators. We also highlight connections between our results and both the
structure of Agler decompositions and study of extreme points for the set of
positive pluriharmonic measures on 2-torus.
