Domain alignment refers broadly to learning correspondences between data distributions from distinct domains. In this work, we focus on a setting where domains share underlying structural patterns despite differences in their specific realizations. The task is particularly challenging in the absence of paired observations, which removes direct supervision across domains. We introduce a generative framework, called SerpentFlow (SharEd-structuRe decomPosition for gEnerative domaiN adapTation), for unpaired domain alignment. SerpentFlow decomposes data within a latent space into a shared component common to both domains and a domain-specific one. By isolating the shared structure and replacing the domain-specific component with stochastic noise, we construct synthetic training pairs between shared representations and target-domain samples, thereby enabling the use of conditional generative models that are traditionally restricted to paired settings. We apply this approach to super-resolution tasks, where the shared component naturally corresponds to low-frequency content while high-frequency details capture domain-specific variability. The cutoff frequency separating low- and high-frequency components is determined automatically using a classifier-based criterion, ensuring a data-driven and domain-adaptive decomposition. By generating pseudo-pairs that preserve low-frequency structures while injecting stochastic high-frequency realizations, we learn the conditional distribution of the target domain given the shared representation. We implement SerpentFlow using Flow Matching as the generative pipeline, although the framework is compatible with other conditional generative approaches. Experiments on synthetic images, physical process simulations, and a climate downscaling task demonstrate that the method effectively reconstructs high-frequency structures consistent with underlying low-frequency patterns, supporting shared-structure decomposition as an effective strategy for unpaired domain alignment.