It is well known that the two-dimensional (2D) nonlinear Schr\"odingerequation (NLSE) with the cubic-quintic (CQ) nonlinearity supports a family ofstable fundamental solitons, as well as solitary vortices (alias vortex rings),which are stable for sufficiently large values of the norm. We study stationarylocalized modes in a symmetric linearly coupled system of two such equations,focusing on asymmetric states. The model may describe "optical bullets" indual-core nonlinear optical waveguides (including spatiotemporal vortices thatwere not discussed before), or a Bose-Einstein condensate (BEC) loaded into a"dual-pancake" trap. Each family of solutions in the single-component model hastwo different counterparts in the coupled system, one symmetric and oneasymmetric. Similarly to the earlier studied coupled 1D system with the CQnonlinearity, the present model features bifurcation loops, for fundamental andvortex solitons alike: with the increase of the total energy (norm), thesymmetric solitons become unstable at a point of the direct bifurcation, whichis followed, at larger values of the energy, by the reverse bifurcationrestabilizing the symmetric solitons. However, on the contrary to the 1Dsystem, the system may demonstrate a double bistability for the fundamentalsolitons. The stability of the solitons is investigated via the computation ofinstability growth rates for small perturbations. Vortex rings, which we studyfor two values of the "spin", s = 1 and 2, may be subject to the azimuthalinstability, like in the single-component model. We also develop aquasi-analytical approach to the description of the bifurcations diagrams,based on the variational approximation. Splitting of asymmetric vortices,induced by the azimuthal instability, is studied by means of directsimulations. Interactions between initially quiescent solitons of differenttypes are studied too.
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