  Inspired by the paper of Bonichon, Bousquet-M\'elou, Dorbec and Pennarun, we
give a system of functional equations which characterise the ordinary
generating function, $U(x),$ for the number of planar Eulerian orientations
counted by edges. We also characterise the ogf $A(x)$, for 4-valent planar
Eulerian orientations counted by vertices in a similar way. The latter problem
is equivalent to the 6-vertex problem on a random lattice, widely studied in
mathematical physics. While unable to solve these functional equations, they
immediately provide polynomial-time algorithms for computing the coefficients
of the generating function. From these algorithms we have obtained 100 terms
for $U(x)$ and 90 terms for $A(x).$
  Analysis of these series suggests that they both behave as $const\cdot (1 -
\mu x)/\log(1 - \mu x),$ where we conjecture that $\mu = 4\pi$ for Eulerian
orientations counted by edges and $\mu=4\sqrt{3}\pi$ for 4-valent Eulerian
orientations counted by vertices.
