Suppose thatXis ap-variate random vector, measured in two populations, with mean vector μjand covariance matrix ψjin populationj,j= 1, 2. In this article we study normal theory estimation of the parameters under the following constraints. It is assumed that (a) the covariance matrices havepidentical eigenvectors, (b) in the coordinate system given by the common eigenvectors, onlyq<pof the means are different between groups, and (c) only theqeigenvalues associated with the sameqcommon eigenvectors are different between groups. There are two main areas of application of these models: (a) in morphometric studies, particularly if size differences between groups are to be removed from the analysis, and (b) in testing for equality of mean vectors and covariance matrices, especially in situations where the number of variablespis large. For applications of type (b), we suggest a randomization test and compare its performance to theT2test and to the likelihood ratio test for equality of both populations.