  In a previous paper, we introduced a new class of Gaussian singular
integrals, that we called the general alternative Gaussian singular integrals
and study the boundedness of them on $L^p(\gamma_d)$, $ 1 < p < \infty.$ In
this paper, we study the boundedness of those operators on Gaussian variable
Lebesgue spaces under a certain additional condition of regularity on
$p(\cdot)$ following a paper by E. Dalmasso and R. Scotto.
