  We construct explicitly non-polynomial eigenfunctions of the difference
operators by Macdonald in case $t=q^k$, $k\in{\mathbb Z}$. This leads to a new,
more elementary proof of several Macdonald conjectures, first proved by
Cherednik. We also establish the algebraic integrability of Macdonald operators
at $t=q^k$ ($k\in {\mathbb Z}$), generalizing the result of Etingof and
Styrkas. Our approach works uniformly for all root systems including $BC_n$
case and related Koornwinder polynomials. Moreover, we apply it for a certain
deformation of $A_n$ root system where the previously known methods do not
work.
