  We study non-Einstein Bach-flat gravitational instanton solutions that can be
regarded as the generalization of the Taub-NUT/Bolt and Eguchi-Hanson solutions
of Einstein gravity to conformal gravity. These solutions include non-Einstein
spaces which are either asymptotically locally flat spacetimes (ALF) or
asymptotically locally Anti-de Sitter (AlAdS). Nevertheless, solutions with
different asymptotic conditions exist: we find geometries that present a
weakened AlAdS asymptotia, exhibiting the typical low decaying mode of
conformal gravity. This permits to identify the simple Neumann boundary
condition that, as it happens in the asymptotically AdS sector, selects the
Einstein solution out of the solutions of conformal gravity. All the geometries
present non-vanishing Hirzebruch signature and Euler characteristic, being
single-centered instantons. We compute the topological charges as well as the
Noether charges of the Taub-NUT/Bolt and Eguchi-Hanson spacetimes, which happen
to be finite. This enables us to study the thermodynamic properties of these
geometries.
