  In the present work we propose a nonlinear anti-$\mathcal{PT}$-symmetric
dimer, that at the linear level has been experimentally created in the realm of
electric circuit resonators. We find four families of solutions, the so-called
upper and lower branches, both in a symmetric and in an asymmetric
(symmetry-broken) form. We unveil analytically and confirm numerically the
critical thresholds for the existence of such branches and explore the
bifurcations (such as saddle-node ones) that delimit their existence, as well
as transcritical ones that lead to their potential exchange of stability. We
find that out of the four relevant branches, only one, the upper symmetric
branch, corresponds to a spectrally and dynamically robust solution. We
subsequently leverage detailed direct numerical computations in order to
explore the dynamics of the different states, corroborating our spectral
analysis results.
