  For the following Ginzburg-Landau system in ${\mathbb R}^2$
  \begin{align*}
  \begin{cases}
  -\Delta w^+
+\Big[A_+\big(|w^+|^2-{t^+}^2\big)+B\big(|w^-|^2-{t^-}^2\big)\Big]w^+=0,
  \\[3mm] -\Delta w^-
+\Big[A_-\big(|w^-|^2-{t^-}^2\big)+B\big(|w^+|^2-{t^+}^2\big)\Big]w^-=0,
\end{cases} \end{align*} with constraints $ A_+, A_->0$, $B<0$, $B^2<A_+A_-$
and $t^+, t^->0$, we will concern its linearized operator ${\mathcal L}$ around
the radially symmetric solution $w(x)=(w^+, w^-): {\mathbb R}^2
\rightarrow\mathbb{C}^2$ of degree pair $(1, 1)$ and prove the non-degeneracy
result: the kernel of ${\mathcal L}$ is spanned by $\big\{\frac{\partial
w}{\partial{x_1}}, \frac{\partial w}{\partial{x_2}}\big\}$ in a natural Hilbert
space. As an application of the non-degeneracy result, a solvability theory for
the linearized operator ${\mathcal L}$ will be given.
