  We consider the problem of finding a maximum popular matching in a
many-to-many matching setting with two-sided preferences and matroid
constraints. This problem was proposed by Kamiyama (2020) and solved in the
special case where matroids are base orderable. Utilizing a newly shown matroid
exchange property, we show that the problem is tractable for arbitrary
matroids. We further investigate a different notion of popularity, where the
agents vote with respect to lexicographic preferences, and show that both
existence and verification problems become coNP-hard, even in the $b$-matching
case.
