Resumen This work addresses the problem of determining whether two multivariate random time series have the same power spectral density(PSD), which has applications, for instance, in physical-layer security and cognitive radio. Remarkably, existing detectors for thisproblem do not usually provide any kind of optimality. Thus, we study here the existence under the Gaussian assumption of optimalinvariant detectors for this problem, proving that the uniformly most powerful invariant test (UMPIT) does not exist. Thus, focusing onclose hypotheses, we show that the locally most powerful invariant test (LMPIT) only exists for univariate time series. In the multivariatecase, we prove that the LMPIT does not exist. However, this proof suggests two LMPIT-inspired detectors, one of which outperformspreviously proposed approaches, as computer simulations show.