  We present a simplified way to construct the Gelfand-Tsetlin modules over
$\mathfrak{gl}(n,\mathbb C)$ related to a $1$-singular GT-tableau defined by
Futorny, Grantcharov and Ramirez. We begin by reframing the classical
construction of generic Gelfand-Tsetlin modules found by Drozd, Futorny and
Ovsienki, showing that they form a flat family over generic points of $\mathbb
C^{\binom{n}{2}}$. We then show that this family can be extended to a flat
family over a variety including generic points and $1$-singular points for a
fixed singular pair of entries. The $1$-singular modules are precisely the
fibers over these points.
