This letter deals with the problem of adaptive detection in partially-homogeneous Gaussian disturbance with unknown but persymmetric structured covariance matrix. Since no uniformly most powerful test exists for the problem at hand, we devise and assess two detection strategies based on the Rao test and the Wald test design criteria. Remarkably, both detectors ensure the constant false alarm rate property with respect to both the structure of the covariance matrix as well as the power level. The preliminary performance assessment, conducted by resorting to simulated data, has confirmed the effectiveness of the newly proposed detectors.