  A topological geon is the quotient manifold $M/Z_2$ where $M$ is a static
spherically symmetric wormhole having the reflection symmetry with respect to
its throat. We distinguish such asymptotically flat solutions of the Einstein
equations according to the form of the time-time metric function by using the
quadrature formulas of the so-called inverse problem for self-gravitating
spherically symmetric scalar fields. We distinguish three types of geon
spacetimes and illustrate them by simple examples. We also study possible
observational effects associated with bounded geodesic motion near topological
geons.
