  We introduce a dual-core system with double symmetry, one between the cores,
and one along each core, imposed by the spatial modulation of local
nonlinearity in the form of two tightly localized spots, which may be
approximated by a pair of ideal delta-functions. The analysis aims to
investigate effects of spontaneous symmetry breaking in such systems.
Stationary one-dimensional modes are constructed in an implicit analytical
form. These solutions include symmetric ones, as well as modes with
spontaneously broken inter-core and along-the-cores symmetries. Solutions
featuring the simultaneous (double) breaking of both symmetries are produced
too. In the model with the ideal delta-functions, all species of the asymmetric
modes are found to be unstable. However, numerical consideration of a two
dimensional extension of the system, which includes symmetric cores with a
nonzero transverse thickness, and the nonlinearity-localization spots of a
small finite size, produces stable asymmetric modes of all the types, realizing
the separate breaking of each symmetry, and states featuring simultaneous
(double) breaking of both symmetries.
