If your graph is $G=(V,E)$ and $B\subseteq E$ are the blue edges, then you can run an appropriate cycle enumeration algorithm on $G-(B\setminus\{ b_i \})$ for each $b_i\in B$. This ensures that at most one blue edge occurs in the graph. Perhaps the selected algorithm can be adapted to start with the edge $b_i$. 

Google reveals many such algorithms. I'm not sure which one suits your needs. No efficient algorithms exist, as along the way you would find Hamiltonian paths, which is an NP-complete problem.