  A doubly group-labeled graph is an oriented graph with its edges labeled by
elements of the direct sum of two groups $\Gamma_1,\Gamma_2$. A cycle in a
doubly group-labeled graph is $(\Gamma_1,\Gamma_2)$-non-zero if it is non-zero
in both coordinates. Our main result is a generalization of the Flat Wall
Theorem of Robertson and Seymour to doubly group-labeled graphs. As an
application, we determine all canonical obstructions to the Erd\H{o}s-P\'osa
property for $(\Gamma_1,\Gamma_2)$-non-zero cycles in doubly group-labeled
graphs. The obstructions imply that the half-integral Erd\H{o}s-P\'osa property
always holds for $(\Gamma_1,\Gamma_2)$-non-zero cycles.
  Moreover, our approach gives a unified framework for proving packing results
for constrained cycles in graphs. For example, as immediate corollaries we
recover the Erd\H{o}s-P\'osa property for cycles and $S$-cycles and the
half-integral Erd\H{o}s-P\'osa property for odd cycles and odd $S$-cycles.
Furthermore, we recover Reed's Escher-wall Theorem.
  We also prove many new packing results as immediate corollaries. For example,
we show that the half-integral Erd\H{o}s-P\'osa property holds for cycles not
homologous to zero, odd cycles not homologous to zero, and $S$-cycles not
homologous to zero. Moreover, the (full) Erd\H{o}s-P\'osa property holds for
$S_1$-$S_2$-cycles and cycles not homologous to zero on an orientable surface.
Finally, we also describe the canonical obstructions to the Erd\H{o}s-P\'osa
property for cycles not homologous to zero and for odd $S$-cycles.
