  Let $M$ be an analytic manifold over $\mathbb{R}$ or $\mathbb{C}$, $\theta$ a
$1$-dimensional Log-Canonical (resp. monomial) singular distribution and
$\mathcal{I}$ a coherent ideal sheaf defined on $M$. We prove the existence of
a resolution of singularities for $\mathcal{I}$ that preserves the
Log-Canonicity (resp. monomiality) of the singularities of $\theta$.
Furthermore, we apply this result to provide a resolution of a family of ideal
sheaves when the dimension of the parameter space is equal to the dimension of
the ambient space minus one.
