  Let $k$ be an arbitrary field. In this note, we show that if a sequence of
relatively prime positive integers ${\bf a}=(a_1,a_2,a_3,a_4)$ defines a
Gorenstein non complete intersection monomial curve ${\mathcal C}({\bf a})$ in
${\mathbb A}_k^4$, then there exist two vectors ${\bf u}$ and ${\bf v}$ such
that ${\mathcal C}({\bf a}+t{\bf u})$ and ${\mathcal C}({\bf a}+t{\bf v})$ are
also Gorenstein non complete intersection affine monomial curves for almost all
$t\geq 0$.
