  We demonstrate that the symmetric elliptic polynomials $E_\lambda(x)$
originally discovered in the study of generalized Noumi-Shiraishi functions are
eigenfunctions of the elliptic Ruijsenaars-Schneider (eRS) Hamiltonians that
act on the mother function variable $y_i$ (substitute of the Young-diagram
variable $\lambda$). This means they are eigenfunctions of the dual eRS system.
At the same time, their orthogonal complements in the Schur scalar product,
$P_R(x)$ are eigenfunctions of the elliptic reduction of the Koroteev-Shakirov
(KS) Hamiltonians. This means that these latter are related to the dual eRS
Hamiltonians by a somewhat mysterious orthogonality transformation, which is
well defined only on the full space of time variables, while the coordinates
$x_i$ appear only after the Miwa transform. This observation explains the
difficulties with getting the apparently self-dual Hamiltonians from the double
elliptic version of the KS Hamiltonians.
