Find maximal subgraph containing only nodes of degree 2 and 3
Basically, in 2-3 graph, the lower bound 2 means that any node has to be in a cycle. 

In DFS, when you reach an already visited node, you found a cycle. Just check if adding this cycle to your subgraph also respect the upper bound 3.

As you suggested, the maximal 2-3 subgraph is not unique. But if you built it following this method, you ensure that any node/vertex of $G$ you did not select in your subgraph:


either is not part of any cycle and would lead to a vertex of degree 1 if selected
either is part of cycles that would all lead to a vertex of degree 4 if selected


Thus, your 2-3 subgraph is maximal.