  Einstein-dilaton-Gauss-Bonnet gravity is investigated on existence of
solutions with mild singularities, not shielded by the event horizons. These
still may have sense since presumably such singularities will be smoothed by
corrections to Einstein theory from quantum gravity/string theory. We show that
gravity with the first-order correction, the Gauss--Bonnet term, gives rise to
special types of singularities, which we call 'cusps', with 1/2-th and 1/3-th
powers in series expansion. The full space then can be split onto several cusps
with classically impenetrable borders, and/or flat asymptotic.
