  A model of the optical media with a spatially structured Kerr nonlinearity is
introduced. The nonlinearity strength is modulated by a set of singular peaks
on top of a self-focusing or defocusing uniform background. The peaks may
include a repulsive or attractive linear potential too. We find that a pair of
mutually symmetric peaks readily gives rise to the spontaneous symmetry
breaking (SSB) of modes pinned to individual peaks, while antisymmetric pinned
modes are always unstable, transforming into robust spatially odd breathers.
Three- and five-peak structures support symmetric modes, with in-phase or
twisted profiles, and do not give rise to asymmetric states. A stability area
is found for the twisted states pinned to the triple peaks, while the
corresponding in-phase modes are unstable, unless the three modulation peaks
are set very close to each other, covered by a single-peak pinned mode. All
patterns pinned to five peaks are unstable too. Collisions of moving solitons
with the singular-modulation peak are studied too. Slowly moving solitons
bounce back from the peak, while the collisions are quasi-elastic for fast
solitons. In the intermediate case, the soliton is destroyed by the collision.
In a special case, the condition of a resonance of the incident soliton with a
trapped mode supported by the peak leads to capture of the soliton.
