  We present a systematic derivation of the abelianity conditions for the
$q$-deformed $W$-algebras constructed from the elliptic quantum algebra
$\mathcal{A}_{q,p}\big(\widehat{\mathfrak{gl}}(N)_{c}\big)$. We identify two
sets of conditions on a given critical surface yielding abelianity lines in the
moduli space ($p, q, c$). Each line is identified as an intersection of a
countable number of critical surfaces obeying diophantine consistency
conditions. The corresponding Poisson brackets structures are then computed for
which some universal features are described.
