  In this work we relate the spectral problem of the toroidal six-vertex
model's transfer matrix with the theory of integrable non-linear differential
equations. More precisely, we establish an analogy between the Classical
Inverse Scattering Method and previously proposed functional equations
originating from the Yang-Baxter algebra. The latter equations are then
regarded as an Auxiliary Linear Problem allowing us to show that the six-vertex
model's spectrum solves Riccati-type non-linear differential equations.
Generating functions of conserved quantities are expressed in terms of
determinants and we also discuss a relation between our Riccati equations and a
stationary Schr\"odinger equation.
