Given a Lax pair $(L(\lambda),M(\lambda))$ i.e a pair of matrices that depends on time $t$ and on a parameter $\lambda$ such that $$\frac{\mathrm{d} L(\lambda)}{\mathrm{d}t} = [M(\lambda), L(\lambda)]$$
We can associate the spectral curve defined by $X_s = \{ (\lambda, \mu) \in \mathbb C^2 : \det(L(\lambda) - \mu) = 0\}$
Eigenvector of $L(\lambda)$ for $\mu$ give a line bundle $\mathcal L\to X_s$, hence a vector field on the Jacobian $J(X_s)$ that varies with $t$. I saw it written several times that if this vector field varies linearly with $t$, it's possible to write explicit solution of the initial equation using theta-functions associated to the Jacobian.
Question : What is a good reference where a simple example (for example, maybe for a $2 \times 2$ matrix, or when the spectral curve is hyperelliptic curve) is worked out in details, with explicit solutions ?
Remarks : Ideally something like this would be nice but I don't understand the physics behind it. Chapter 2 of this reference is also quite nice but they don't talk about the Jacobian. Examples like geodesics flows on ellipsoid look beautiful but are maybe a bit involved.