In this chapter, we introduce Lax pairs with spectral parameters. These are Lax matrices L(λ) and M(λ) depending analytically on a parameter λ. The study of the analytical properties of the Lax equation L(λ) = [M(λ), L(λ)] yields considerable insight into its structure, and in fact, quickly introduces many of the major objects and concepts, which will be developed in depth in the subsequent chapters.

The first important result is that the possible forms of M(λ) are completely determined by eq. (3.15). This form of M(λ) is such that the commutator [M(λ), L(λ)] has the same polar structure as L(λ). The Lax equation has then a natural interpretation as a flow on a coadjoint orbit of a loop group. This has in turn the important consequence of introducing a symplectic structure into the theory allowing us to connect with Liouville integrability. Moreover, this geometric interpretation of the Lax equation lends itself to its solution by factorization in a loop group, which is a Riemann—Hilbert problem. Studying the analytic structure of M(λ), we are led to consider an infinite family of elementary flows, depending on the order of the poles. This introduces a connection between time flows and the spectral parameter dependence, which finds a striking expression in Sato's formula expressing the wave function in terms of tau-functions, eq. (3.61).