In the general framework of radar detection, estimation of the Gaussian or non-Gaussian clutter covariance matrix is an important point. This matrix commonly exhibits a particular structure: for instance, this is the case for active systems using a symmetrically spaced linear array with constant pulse repetition interval. We propose using the particular persymmetric structure of the covariance matrix to improve the detection performance. In this context, this work provides two new adaptive detectors for Gaussian additive noise and non-Gaussian additive noise which is modeled by the spherically invariant random vector (SIRV). Their statistical properties are then derived and compared with simulations. The vast improvement in their detection performance is demonstrated by way of simulations or experimental ground clutter data. This allows for the analysis of the proposed detectors on both real Gaussian and non-Gaussian data.