  We consider the extension of the thermodynamic Bethe Ansatz (TBA) to cases in
which additional terms involving higher conserved charges are added to the
Hamiltonian, or in which a distinction is made between the Hamiltonian used for
time evolution and that used for defining the density matrix. Writing down
equations describing the saddle-point (pseudo-equilibrium) state of the
infinite system, we prove the existence and uniqueness of solutions for
Lieb-Liniger provided simple requirements are met. We show how a knowledge of
the saddle-point rapidity distribution is equivalent to that of all generalized
chemical potentials, and how the standard equilibrium equations for e.g.
excitations are simply generalized.
