  Nurowski showed that any generic 2-plane field $D$ on a 5-manifold $M$
determines a natural conformal structure $c_D$ on $M$; these conformal
structures are exactly those (on oriented $M$) whose normal conformal holonomy
is contained in the (split, real) simple Lie group $G_2$. Graham and Willse
showed that for real-analytic $D$ the same holds for the holonomy of the
real-analytic Fefferman-Graham ambient metric of $c_D$, and that both holonomy
groups are equal to $G_2$ for almost all $D$. We investigate here independently
interesting plane fields for which the associated holonomy groups are a proper
subset of $G_2$.
  Cartan solved the local equivalence problem for $2$-plane fields $D$ and
constructed the fundamental curvature tensor $A$ for these objects. He
furthermore claimed to describe locally all $D$ whose infinitesimal symmetry
algebra has rank at least $6$ and gave a local quasi-normal form, depending on
a single function of one variable, for those that furthermore satisfy a natural
degeneracy condition on $A$, but Doubrov and Govorov recently rediscovered a
counterexample to Cartan's claim. We show that for all $D$ given by Cartan's
alleged quasi-normal form, the conformal structures $c_D$ induced via
Nurowski's construction are almost Einstein, that we can write their ambient
metrics explicitly, and that the holonomy groups associated to $c_D$ are always
the $5$-dimensional Heisenberg group, which here acts indecomposably but not
irreducibly. (Not all of these properties hold, however, for Doubrov and
Govorov's counterexample.) We also show that the similar results hold for the
related class of $2$-plane fields defined on suitable jet spaces by ordinary
differential equations $z'(x) = F(y''(x))$ satisfying a simple genericity
condition.
