Adaptive detection of a signal competing with Gaussian interference with spatially varying power is addressed in this letter. Invariance principle is used to focus the attention on a class of decision rules exhibiting some specific symmetries, which represent a sufficient condition to ensure the constant false alarm rate property. The class of invariant decision rules is characterized via the design of a maximal invariant statistic together with its equivalent in the parameter space. Finally, it is proved that the adaptive normalized matched filter using a suitable recursive estimate of the interference covariance structure coincides with a maximal invariant component.