  Much attention has been recently devoted to modified theories of gravity in
the attempt to efficiently describe both early inflation and late-time
acceleration of our universe without referring to the cosmological constant or
other ad hoc kinds of fluids. The simplest models overcome General Relativity
simply by replacing $R$ with $F(R)$ in the Einstein--Hilbert action.
Unfortunately, such models typically lack most of the beautiful solutions
discovered in Einstein's gravity. Nonetheless, in $F(R)$ gravity, it has been
possible to get at least few black holes, but still we do not know any empty
wormhole-like spacetime solution. The present paper aims to explain why it is
so hard to get such solutions (given that they exist). Few solutions are
derived in the simplest cases while only an implicit form has been obtained in
the non-trivial case.
