  We construct new solutions of five-dimensional quadratic gravity as direct
products of a constant curvature two-surface with a solution of
three-dimensional new massive gravity with constant scalar curvature. These
solutions could represent near-horizon limits of five-dimensional
asymptotically flat black strings or black rings. A number of these
non-asymptotically flat solutions are themselves black strings or rings. A
by-product of our analysis is the construction of new solutions of
four-dimensional quadratic gravity obtained by toroidal reduction of the
five-dimensional solutions with flat transverse space. These again include
black strings or rings, and an $AdS_2\times T^2$ solution of $f(R)$ gravity for
a specific relation between the model parameters.
