  The BT of Adler's lattice equation is inherent in the equation itself by
virtue of its multidimensional consistency. We refer to a solution of the
equation that is related to itself by the composition of two BTs (with
different Backlund parameters) as a 2-cycle of the BT. In this article we will
show that such solutions are associated with a commuting one-parameter family
of rank-2 (i.e., 2-variable), 2-valued mappings. We will construct the explicit
solution of the mappings within this family and hence give the solutions of
Adler's equation that are 2-cycles of the BT.
