Black ringoids: spinning balanced black objects in $d\geq 5$ dimensions -- the codimension-two case

We propose a general framework for the study of asymptotically flat black objects with $k+1$ equal magnitude angular momenta in $d\geq 5$ spacetime dimensions (with $0\leq k\leq \big[\frac{d-5}{2} \big]$). In this approach, the dependence on all angular coordinates but one is factorized, which leads to a codimension-two problem. This framework can describe black holes with spherical horizon topology, the simplest solutions corresponding to a class of Myers-Perry black holes. A different set of solutions describes balanced black objects with $S^{n+1} \times S^{2k+1}$ horizon topology. The simplest members of this family are the black rings $(k=0)$. The solutions with $k>0$ are dubbed $black~ringoids$. Based on the nonperturbative numerical results found for several values of $(n,k)$, we propose a general picture for the properties and the phase diagram of these solutions and the associated black holes with spherical horizon topology: $n=1$ black ringoids repeat the $k=0$ pattern of black rings and Myers-Perry black holes in 5 dimensions, whereas $n>1$ black ringoids follow the pattern of higher dimensional black rings associated with `pinched' black holes and Myers-Perry black holes.