  We prove that the regular generalized cluster structure on the Drinfeld
double of $GL_n$ constructed in arXiv:1912.00453 is complete and compatible
with the standard Poisson--Lie structure on the double. Moreover, we show that
for $n=4$ this structure is distinct from a previously known regular
generalized cluster structure on the Drinfeld double, even though they have the
same compatible Poisson structure and the same collection of frozen variables.
Further, we prove that the regular generalized cluster structure on band
periodic matrices constructed in arXiv:1912.00453 possesses similar
compatibility and completeness properties.

Representation Theory;Commutative Algebra;Quantum Algebra