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In this paper we consider adaptive detection of a signal embedded in additive disturbance whose multivariate distribution belongs to a very general class, including many statistical models commonly adopted for radar disturbance. We introduce the concept of generalized Constant False Alarm Rate (CFAR) and show that a class of receivers sharing some invariances complies with the quoted property. Then, we devise the Generalized Likelihood Ratio Test (GLRT) and prove that, under some mild technical conditions, it coincides with that obtained under the Gaussian assumption for the observations. We also deal with the existence of the Uniformly Most Powerful Invariant (UMPI) detector either using the Wijsman theorem or directly computing the maximal invariant Likelihood Ratio (LR). At the analysis stage, we focus on a compound matrix variate model for the disturbance component, which is a natural generalization of the Spherically Invariant Random Vector (SIRV). In this context, we assess the performance of some well known invariant decision rules also in comparison with the Most Powerful Invariant (MPI) detector. The results highlight that some among the analyzed receivers exhibit a performance level very close to the MPI test.