  Mobiles are a particular class of decorated plane trees which serve as
codings for planar maps. Here we address the question of enumerating mobiles in
their most general flavor, in correspondence with planar Eulerian (i.e.,
bicolored) maps. We show that the generating functions for such mobiles satisfy
a number of recursive equations which lie in the field of integrable systems,
leading us to explicit expressions for these generating functions as ratios of
particular determinants. In particular we recover known results for mobiles
associated with uncolored maps and prove some conjectured formulas for the
generating functions of mobiles associated with $p$-constellations.
