  Let $X$ be a connected non-compact $2$-dimensional manifold possibly with
boundary and $\Delta$ be a foliation on $X$ such that each leaf
$\omega\in\Delta$ is homeomorphic to $\mathbb{R}$ and has a trivially foliated
neighborhood. Such foliations on the plane were studied by W. Kaplan who also
gave their topological classification. He proved that the plane splits into a
family of open strips foliated by parallel lines and glued along some boundary
intervals. However W. Kaplan's construction depends on a choice of those
intervals, and a foliation is described in a non-unique way. We propose a
canonical cutting by open strips which gives a uniqueness of classifying
invariant. We also describe topological types of closures of those strips under
additional assumptions on $\Delta$.
