Comments: Title changed, references added, matches the published version
We study a generalized QFT of the Coulomb branch in 6d $SU(N)_k$ gauge theories and show that the space of integrable extensions is fully finite in the Coulomb branch. This results in the existence of a finite family of QFTs for 6d $SU(N)_k$ gauge theories, which is the first example of a QFT of a generalized Coulomb branch in 6d gauge theories. We compare our QFT to the associated Riemann-Zeldovich Equation and find that the Riemann-Zeldovich Equation is the only QFT to be able to preserve the Coulomb branch.