Cycle double covers and non-separating cycles

Which 2-regular subgraph R of a cubic graph G can be extended to a cycle double cover of G? We provide a condition which ensures that every R satisfying this condition is part of a cycle double cover of G. As one consequence, we prove that every 2-connected cubic graph which has a decomposition into a spanning tree and a 2-regular subgraph C consisting of k circuits with k≤3, has a cycle double cover containing C.