Decomposition of infinite eulerian graphs with a small number of vertices of infinite degree

We consider the question whether an infinite eulerian graph has a decomposition into circuits and rays if the graph has only finitely many, say n, vertices of infinite degree, and only finitely many finite components after the removal of the vertices of infinite degree. It is known that the answer is affirmative for n2 and negative for n4. We settle the remaining case n=3, showing that a decomposition into circuits and rays also exists in this case.