This paper deals with the problem of adaptive multidimensional/multichannel signal detection in homogeneous Gaussian disturbance with unknown covariance matrix and structured deterministic interference. The aforementioned problem corresponds to a generalization of the well-known Generalized Multivariate Analysis of Variance (GMANOVA). In this first part of the work, we formulate the considered problem in canonical form and, after identifying a desirable group of transformations for the considered hypothesis testing, we derive a Maximal Invariant Statistic (MIS) for the problem at hand. Furthermore, we provide the MIS distribution in the form of a stochastic representation. Finally, strong connections to the MIS obtained in the open literature in simpler scenarios are underlined.