  We construct a basis of solutions of the scalar $\boldsymbol{ \texttt{t} }-
\boldsymbol{ \texttt{Q} }$ equation describing the spectrum of the $q$-Toda and
Toda$_2$ chains by using auxiliary non-linear integral equations. Our
construction allows us to provide quantisation conditions for the spectra of
these models in the form of thermodynamic Bethe Ansatz-like equations.
