The set of all normal surfaces in a 3-manifold is a partial monoid under addition with a minimal generating set of fundamental surfaces. The available algorithm for finding the system of fundamental surfaces is of a theoretical nature and admits no implementation in practice. In this article, we give a complete and geometrically simple description for the structure of partial monoids for normal surfaces in lens spaces, generalized quaternion spaces, and Stallings manifolds with fiber a punctured torus and a hyperbolic monodromy map.