  We study two-dimensional holomorphic distributions on $\mathbb{P}^4$. We
classify dimension two distributions, of degree at most $2$, with either
locally free tangent sheaf or locally free conormal sheaf and whose singular
scheme has pure dimension one. We show that the corresponding sheaves are
split. Next, we investigate the geometry of such distributions, studying from
maximally non-integrable to integrable distributions. In the maximally
non-integrable case, we show that the distribution is either of Lorentzian type
or a push-forward by a rational map of the Cartan prolongation of a singular
contact structure on a weighted projective 3-fold. We study distributions of
dimension two in $\mathbb{P}^4$ whose the conormal sheaves are the
Horrocks-Mumford sheaves, describing the numerical invariants of their singular
schemes which are smooth and connected. Such distributions are maximally
non-integrable, uniquely determined by their singular schemes and invariant by
a group $H_5 \rtimes SL(2,\mathbb{Z}_5) \subset Sp(4, \mathbb{Q})$, where $H_5$
is the Heisenberg group of level $5$. We prove that the moduli spaces of
Horrocks-Mumford distributions are irreducible quasi-projective varieties and
we determine their dimensions. Finally, we observe that the space of
codimension one distributions, of degree $d\geq 6$, on $\mathbb{P}^4$ have a
family of degenerated flat holomorphic Riemannian metrics. Moreover, the
degeneracy divisors of such metrics consist of codimension one distributions
invariant by $H_5 \rtimes SL(2,\mathbb{Z}_5)$ and singular along a degenerate
abelian surface with $(1,5)$-polarization and level-$5$-structure.
