  Let $G=(G_{1},G_{2})$ be a Gaussian vector in $R^{2}$ with $EG_{1}G_{2}\neq
0$. Let $c_{1},c_{2}\in R^{1}$. A necessary and sufficient condition for
$G=((G_{1}+c_{1}\alpha)^{2},(G_{2}+c_{2}\alpha)^{2})$ to be infinitely
divisible for all $\alpha\in R^{1}$ is that \[ \Ga_{i,i}\geq
\frac{c_{i}}{c_{j}}\Ga_{i,j}>0\qquad\forall 1\le i\ne j\le 2.\] In this paper
we show that when this does not hold there exists an $0<\alpha_{0}<\ff $ such
that $G=((G_{1}+c_{1}\alpha)^{2},(G_{2}+c_{2}\alpha)^{2})$ is infinitely
divisible for all $|\alpha|\leq \alpha_{0}$ but not for any $|\al|>\al_{0}$.
