We present a first two-dimensional general relativistic analysis of superconducting regions in axially symmetric highly magnetized neutron star (magnetar) models with toroidal magnetic fields. We investigate the topology and distribution of type-II and type-I superconducting regions for varying toroidal magnetic field strengths and stellar masses by solving the Einstein-Maxwell equations using the xns code. Our results reveal that the outer cores of low- to intermediate-mass magnetars sustain superconductivity over larger regions compared to higher-mass stars with nontrivial distribution of type-II and type-I regions. Consistent with previous one-dimensional (1D) models, we find that regardless of the gravitational mass, the inner cores of magnetars with toroidal magnetic fields are devoid of $S$-wave proton superconductivity. Furthermore, these models contain nonsuperconducting, torus-shaped regions; a novel feature absent in previous 1D studies. Finally, we speculate on the potential indirect effects of superconductivity on continuous gravitational wave emissions from millisecond pulsars, such as PSR J1843-1113, highlighting their relevance for future gravitational wave detectors.

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• Magnetic vortices
• Superconductivity
• Neutron stars & pulsars
• Stellar interior