  This paper is about geometric and Riemannian properties of Engel structures,
i.e. maximally non-integrable $2$-plane fields on $4$-manifolds. Two $1$-forms
$\alpha$ and $\beta$ are called Engel defining forms if
$\mathcal{D}=\ker\alpha\cap\ker\beta$ is an Engel structure and
$\mathcal{E}=\ker\alpha$ is its associated even contact structure, i.e.
$\mathcal{E}=[\mathcal{D},\mathcal{D}]$. A choice of Engel defining forms
determines a distribution $\mathcal{R}$ transverse to $\mathcal{D}$ called the
Reeb distribution. We study conditions that ensure integrability of
$\mathcal{R}$. For example if we have a metric $g$ which makes the splitting
$TM=\mathcal{D}\oplus\mathcal{R}$ orthogonal and such that $\mathcal{D}$ is
totally geodesic then there exists an integrable Reeb distribution
$\tilde{\mathcal{R}}$.
  It turns out that integrabilty of $\mathcal{R}$ is related to the existence
of vector fields $Z$ whose flow preserves $\mathcal{D}$, so called Engel vector
fields. A K-Engel structure is a triple $(\mathcal{D},\,g,\,Z)$ where
$\mathcal{D}$ is an Engel structure, $g$ is a Riemannian metric, and $Z$ is a
vector field which is Engel, Killing, and orthogonal to $\mathcal{E}$. In this
case we can construct Engel defining forms with very nice properties and such
that $\mathcal{R}$ is integrable. Moreover we can classify the topology of
K-Engel manifolds studying the action of the flow of $Z$. As natural
consequences of these methods we provide a construction which is the analogue
of the Boothby-Wang construction in the contact setting and we give a notion of
contact filling for an Engel structure.
