  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 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.
