One of the biggest successes in the theory of (classical) integrable systems is that models that possess a Lax pair and are amenable to the inverse scattering method turn out to also possess a Hamiltonian description (several in fact). The connection between these two aspects is at the core of the theory and is well documented. Each aspect allows one to study different problems: geometric structure (Hamiltonian) and solution content (Lax pair). The question of nontrivial boundary conditions and the construction of solutions in such models was first considered as early as 1975 by Ablowitz and Segur but only from Lax pair point of view. In 1987, Sklyanin's seminal work laid the foundations to define and study integrable boundary conditions from both the Hamiltonian and Lax pair point of views. Both aspects seem to have developed independently though and the following points were never addressed properly: 1) Can one derive the Lax pair point of view from the Hamiltonian one in a way similar to what is known in the case without boundary conditions? 2) How can one understand the apparent gap between the two approaches that predict different integrable boundary conditions?

In a recent work with J. Avan and N. Crampé, we addresses and solved these 30 year-old problems. More recently, with N. Crampé, we used the example of the Ablowitz-Ladik model to illustrate the construction and obtained new integrable boundary conditions. We also implemented the entire framework that goes from Hamiltonian description to the explicit construction of soliton solutions for the problem on the half-line with these new boundary conditions.

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