On the possibility of constructing a curve for a given group of homeomorphisms

The following problem is considered: for a given group of homeomorphisms of a topological space, it is required to determine if there exists in this space a curve for which the given group is a group of oriented homeomorphisms. A constructive solution of the problem is given for a wide class of groups of homeomorphisms of linearly connected topological spaces. In a number of cases, the questions on the uniqueness of the constructed curve and on the kernel of action of the group on the curve are investigated.