  The properties of static, spherically symmetric configurations are considered
in the framework of two models of nonlocally corrected gravity, suggested in S.
Deser and R. Woodard., Phys. Rev. Lett. 663, 111301 (2007), and S. Capozziello
et al., Phys. Lett. B 671, 193 (2009). For the first case, where the Lagrangian
of nonlocal origin represents a scalar-tensor theory with two massless scalars,
an explicit condition is found under which both scalars are canonical
(non-phantom). If this condition does not hold, one of the fields exhibits a
phantom behavior. Scalar-vacuum configurations then behave in a manner known
for scalar-tensor theories. In the second case, the Lagrangian of nonlocal
origin exhibits a scalar field interacting with the Gauss-Bonnet (GB) invariant
and contains an arbitrary scalar field potential. It is found that the GB term,
in general, leads to violation of the well-known no-go theorems valid for
minimally coupled scalar fields in general relativity. It is shown, however,
that some configurations of interest are still forbidden, whatever be the
scalar field potential and the GB-scalar coupling function, namely,
"force-free" wormholes (such that g_{tt}= const) and black holes with
higher-order horizons.
