  We extend our theory of slow magnons in a two-component Bose-Einstein
condensate to the case of two spatial dimensions (2D). We provide a detailed
discussion of polaronic corrections to the magnon branch of the elementary
excitation spectrum in a weakly- and strongly-interacting regimes. In a dilute
system, the latter may be achieved by adjusting inter-species attraction such
as to obtain a resonance in the $p$-wave scattering channel. Resonantly
enhanced $p$-wave attraction results in the self-localization of magnons and
formation of a magnetoroton. The nature of a ground state beyond the
magnetoroton instability remains to be explored. We suggest a dilute p-wave
crystal of alternating polarization as a possible candidate. In contrast to
three dimensions (3D), the required strength of the attractive potential in the
$s$-wave channel here corresponds to tight binding, and we suggest a potential
realization of our model with excitons in 2D semiconductors.
