  The following problem is addressed: A $3$-manifold $M$ is endowed with a
triple $\Omega = \big(\Omega^1,\Omega^2,\Omega^3\big)$ of closed $2$-forms. One
wants to construct a coframing $\omega = \big(\omega^1,\omega^2,\omega^3\big)$
of $M$ such that, first, ${\rm d}\omega^i = \Omega^i$ for $i=1,2,3$, and,
second, the Riemannian metric
$g=\big(\omega^1\big)^2+\big(\omega^2\big)^2+\big(\omega^3\big)^2$ be flat. We
show that, in the 'nonsingular case', i.e., when the three $2$-forms
$\Omega^i_p$ span at least a $2$-dimensional subspace of $\Lambda^2(T^*_pM)$
and are real-analytic in some $p$-centered coordinates, this problem is always
solvable on a neighborhood of $p\in M$, with the general solution $\omega$
depending on three arbitrary functions of two variables. Moreover, the
characteristic variety of the generic solution $\omega$ can be taken to be a
nonsingular cubic. Some singular situations are considered as well. In
particular, we show that the problem is solvable locally when $\Omega^1$,
$\Omega^2$, $\Omega^3$ are scalar multiples of a single 2-form that do not
vanish simultaneously and satisfy a nondegeneracy condition. We also show by
example that solutions may fail to exist when these conditions are not
satisfied.
