  We extend our study of integrable structures in large $N_c$ QCD$_2$ to a
broad class of theories called the generalized QCD, which are given by the
Lagrangian $\mathcal{L}\propto {\rm tr}\,B\wedge F- {\rm tr}\,V(B)$ coupled to
quarks in the fundamental representation. We recast the Bethe-Salpeter equation
for the meson spectrum into a TQ-Baxter equation and determine a transfer
matrix in a closed form for any given polynomial $V(B)$. Using this
reformulation, we derive the asymptotic expansion of energy levels and
wavefunctions. Examining the analytic structure of the spectrum as a function
of the coefficients of $V(B)$, we determine the region of couplings where the
theory admits a positive and discrete spectrum of mesons. Furthermore, we
uncover a multi-sheeted structure with infinitely many multi-critical points,
where several mesons become simultaneously massless. Lastly, we illustrate that
this structure persists in the large-representation limit of the generalized
QCD with the SU(2) gauge group.
