A map is unicursal if all its vertices are even-valent except two odd-valent vertices. A near-triangulation is a map with all but one of its face triangles. This paper investigates the enumeration of rooted unicursal planar near-triangulation and provides two functional equations satisfied by its generating functions with the root-face valency, the number of edges and the number of nonrooted vertices of maps as parameters.

unicursal map, near-triangulation, enumerating function, functional equation.