  The parity of the length of paths and cycles is a classical and well-studied
topic in graph theory and theoretical computer science. The parity constraints
can be extended to label constraints in a group-labeled graph, which is a
directed graph with each arc labeled by an element of a group. Recently, paths
and cycles in group-labeled graphs have been investigated, such as packing
non-zero paths and cycles, where "non-zero" means that the identity element is
a unique forbidden label.
  In this paper, we present a solution to finding an $s$--$t$ path with two
labels forbidden in a group-labeled graph. This also leads to an elementary
solution to finding a zero $s$--$t$ path in a ${\mathbb Z}_3$-labeled graph,
which is the first nontrivial case of finding a zero path. This situation in
fact generalizes the 2-disjoint paths problem in undirected graphs, which also
motivates us to consider that setting. More precisely, we provide a
polynomial-time algorithm for testing whether there are at most two possible
labels of $s$--$t$ paths in a group-labeled graph or not, and finding $s$--$t$
paths attaining at least three distinct labels if exist. The algorithm is based
on a necessary and sufficient condition for a group-labeled graph to have
exactly two possible labels of $s$--$t$ paths, which is the main technical
contribution of this paper.
