  We introduce the notion of $\varepsilon\eta\,$-Einstein
$\varepsilon\,$-contact metric three-manifold, which includes as particular
cases $\eta\,$-Einstein Riemannian and Lorentzian (para) contact metric
three-manifolds, but which in addition allows for the Reeb vector field to be
null. We prove that the product of an $\varepsilon\eta\,$-Einstein Lorentzian
$\varepsilon\,$-contact metric three-manifold with an
$\varepsilon\eta\,$-Einstein Riemannian contact metric three-manifold carries a
bi-parametric family of Ricci-flat Lorentzian metric-compatible connections
with isotropic, totally skew-symmetric, closed and co-closed torsion, which in
turn yields a bi-parametric family of solutions of six-dimensional minimal
supergravity coupled to a tensor multiplet. This result allows for the
systematic construction of families of Lorentzian solutions of six-dimensional
supergravity from pairs of $\varepsilon\eta\,$-Einstein contact metric
three-manifolds. We classify all left-invariant $\varepsilon\eta\,$-Einstein
structures on simply connected Lie groups, paying special attention to the case
in which the Reeb vector field is null. In particular, we show that the Sasaki
and K-contact notions extend to $\varepsilon\,$-contact structures with null
Reeb vector field but are however not equivalent conditions, in contrast to the
situation occurring when the Reeb vector field is not light-like. Furthermore,
we pose the Cauchy initial-value problem of an $\varepsilon\,$-contact
$\varepsilon\eta\,$-Einstein structure, briefly studying the associated
constraint equations in a particularly simple decoupling limit. Altogether, we
use these results to obtain novel families of six-dimensional supergravity
solutions, some of which can be interpreted as continuous deformations of the
maximally supersymmetric solution on
$\widetilde{\mathrm{Sl}}(2,\mathbb{R})\times S^3$.
