  We address the six vertex model on a random lattice, which in combinatorial
terms corresponds to the enumeration of weighted 4-valent planar maps equipped
with an Eulerian orientation. This problem was exactly, albeit non-rigorously
solved by Ivan Kostov in 2000 using matrix integral techniques. We convert
Kostov's work to a combinatorial argument involving functional equations coming
from recursive decompositions of the maps, which we solve rigorously using
complex analysis. We then investigate modular properties of the solution, which
lead to simplifications in certain special cases. In particular, in two special
cases of combinatorial interest we rederive the formulae discovered by
Bousquet-M\'elou and the first author.
