  In this work we study families of pairs of window functions and lattices
which lead to Gabor frames which all possess the same frame bounds. To be more
precise, for every generalized Gaussian $g$, we will construct an uncountable
family of lattices $\lbrace \Lambda_\tau \rbrace$ such that each pairing of $g$
with some $\Lambda_\tau$ yields a Gabor frame, and all pairings yield the same
frame bounds. On the other hand, for each lattice we will find a countable
family of generalized Gaussians $\lbrace g_i \rbrace$ such that each pairing
leaves the frame bounds invariant. Therefore, we are tempted to speak about
"Gabor Frame Sets of Invariance".
