  We address the double bubble problem for the anisotropic Grushin perimeter
$P_\alpha$, $\alpha\geq 0$, and the Lebesgue measure in $\mathbb R^2$, in the
case of two equal volumes. We assume that the contact interface between the
bubbles lays on either the vertical or the horizontal axis. Since no regularity
theory is available in this setting, in both cases we first prove existence of
minimizers via the direct method by symmetrization arguments and then
characterize them in terms of the given area by first variation techniques.
Angles at which minimal boundaries intersect satisfy the standard 120-degree
rule up to a suitable change of coordinates. While for $\alpha=0$ the Grushin
perimeter reduces to the Euclidean one and both minimizers coincide with the
symmetric double bubble found in [Foisy et al., Pacific J. Math. (1993)], for
$\alpha=1$ vertical interface minimizers have Grushin perimeter strictly
greater than horizontal interface minimizers. As the latter ones are obtained
by translating and dilating the Grushin isoperimetric set found in [Monti
Morbidelli, J. Geom. Anal. (2004)], we conjecture that they solve the double
bubble problem with no assumptions on the contact interface.
