  The recent creation of Townes solitons (TSs) in binary Bose-Einstein
condensates and experimental demonstration of spontaneous symmetry breaking
(SSB) in solitons propagating in dual-core optical fibers draw renewed interest
to the TS and SSB phenomenology in these and other settings. In particular,
stabilization of TSs, which are always unstable in free space, is a relevant
problem with various ramifications. We introduce a system which admits exact
solutions for both TSs and SSB of solitons. It is based on a dual-core
waveguide with quintic self-focusing and fused (localized) coupling between the
cores. The respective system of coupled nonlinear Schroedinger equations gives
rise to exact solutions for full families of symmetric solitons and asymmetric
ones, which are produced by the supercritical SSB bifurcation (i.e., the
symmetry-breaking phase transition of the second kind). Stability boundaries of
asymmetric solitons are identified by dint of numerical methods. Unstable
solitons spontaneously transform into robust moderately asymmetric breathers or
strongly asymmetric states with small intrinsic oscillations. The setup can be
used in the design of photonic devices operating in coupling and switching
regimes.
