Relativistic elliptic matrix tops and finite Fourier transformations

We consider a family of classical elliptic integrable systems including (relativistic) tops and their matrix extensions of different types. These models can be obtained from the "off-shell" Lax pairs, which do not satisfy the Lax equations in general case but become true Lax pairs under various conditions (reductions). At the level of the off-shell Lax matrix, there is a natural symmetry between the spectral parameter z and relativistic parameter eta. It is generated by the finite Fourier transformation, which we describe in detail. The symmetry allows one to consider z and eta on an equal footing. Depending on the type of integrable reduction, any of the parameters can be chosen to be the spectral one. Then another one is the relativistic deformation parameter. As a by-product, we describe the model of N-2 interacting GL(M) matrix tops and/or M-2 interacting GL(N) matrix tops depending on a choice of the spectral parameter.