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There are some integrable Lotka--Volterra systems, see e.g. the paper O.I. Bogoyavlenskij, Integrable Lotka--Volterra systems, Regular and Chaotic Dynamics, 2008, Vol. 13, No. 6, pp. 543–556.

There are such examples (with transcendental integrals of motion) already on $\mathbb{R}^{4}$, see the paper of Hietarinta. The Hamiltonian is $$H=p_x^2/2+p_y^2/2+2y p_x p_y-x,$$ the desired integra …

The point is that you should assume that $a,b,c$ are polynomials in $\lambda$ and then equate to zero the coefficients at various powers of $\lambda$. Similar analysis can, I think, be found in many b …

Your understanding is essentially correct. There are three basic (and closely related) approaches to constructing the integrals of motion required for complete integrability: through separation of var …

Regarding 2), I'm not an expert in algebraic integrable systems, but shouldn't it be the other way around? That is, given an integrable evolutionary system of PDEs which is further assumed to be Hamil …

As for $I_4$ and $I_j$ for all $j\geqslant 4$ up to overall sign factors, see e.g. equation (1.9) of these lecture notes with $u=-r$. Also you may wish to look at this link. Regarding the bracket at …

In brief, an integrable hierarchy is an infinite (usually countable) set of integrable partial differential systems such that any two systems in this set are compatible. Such hierarchies are usually g …

The above answers deal mostly with finite-dimensional systems. As for the (systems of) PDEs, you typically need the Lax pair or a zero curvature representation (see e.g. the Takhtajan--Faddeev book me …