Two-dimensional Toda field equations related to the exceptional algebra $\mathfrak g_2$: Spectral properties of the Lax operators

We analyze spectral properties of the Lax operator corresponding to the two-dimensional Toda field equations related to the algebra $\mathfrak g_2$. We construct two minimal sets of scattering data $\mathcal T_s$, $s=1,2$, understanding the map between the potential and each of the sets $\mathcal T_s$ as a generalized Fourier transformation. We construct explicit recursion operators with special factorization properties.