  A nonlocal gravity model (2.1) was introduced and considered recently [49],
and two exact cosmological solutions in flat space were presented. The first
solution is related to some radiation effects generated by nonlocal dynamics on
dark energy background, while the second one is a nonsingular time symmetric
bounce. In the present paper we investigate other possible exact cosmological
solutions and find some the new ones in nonflat space. Used nonlocal gravity
dynamics can change background topology. To solve the corresponding eqations of
motion, we first look for a solution of the eigenvalue problem $\Box (R
-4\Lambda) = q\ (R - 4\Lambda) .$ We also discuss possible extension of this
model with nonlocal operator symmetric under $\Box \longleftrightarrow
\Box^{-1}$ and its connection with another interesting nonlocal gravity model.
