Let there be given a $2n$-dimenional real symplectic manifold $(M,\omega)$ with a globally defined real function $H:M\times[t_i,t_f] \to \mathbb{R}$, which we will call the Hamiltonian. The time evolution is governed by Hamilton's (or equivalently Liouville's) equations of motion. Here $t\in[t_i,t_f]$ is time.
On one hand, there is the notion of complete integrability, aka. Liouville integrability, or sometimes just called integrability. This means that there exist $n$ independent globally defined real functions
$$I_i, \qquad i\in\{1, \ldots, n\},$$
(which we will call action variables), that pairwise Poisson commute,
$$ \{I_i,I_j\}_{PB}~=~0, \qquad i,j\in\{1, \ldots, n\}.$$
On the other hand, given a fixed point $x_{(0)}\in M$, under mild regularity assumptions, there always exists locally (in a sufficiently small open Darboux$^1$ neighborhood of $x_{(0)}$) an $n$-parameter complete solution for Hamilton's principal function
$$S(q^1, \ldots, q^n; I_1, \ldots,I_n; t)$$
to the Hamilton-Jacobi equation, where
$$I_i, \qquad i\in\{1, \ldots, n\},$$
are integration constants. This leads to a local version of property 1.
The main point is that the global property 1 is rare, while the local property 2 is generic.
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$^1$ A Darboux neighborhood here means a neighborhood where there exists a set of canonical coordinates aka. Darboux coordinates $(q^1, \ldots, q^n;p_1, \ldots, p_n)$, cf. Darboux' Theorem.
1) A constant of motion
$f(z,t)$ is a (globally defined, smooth) function $f:M\times [t_i,t_f] \to
\mathbb{R}$ of the dynamical variables $z\in M$ and time $t\in[t_i,t_f]$,
such that the map $$[t_i,t_f]~\ni ~t~~\mapsto~~f(\gamma(t),t)~\in~ \mathbb{R}$$
doesn't depend on time for every solution curve $z=\gamma(t)$ to the equations of motion of the system.
An integral of motion/first integral
is a constant of motion $f(z)$ that doesn't depend explicitly on time.
2) In the following let us for simplicity restrict to the case where the system is a finite-dimensional autonomous$^1$ Hamiltonian system with Hamiltonian $H:M \to \mathbb{R}$ on a $2N$-dimensional symplectic manifold $(M,\omega)$.
Such system is called (Liouville/completely) integrable if
there exist $N$ functionally independent$^2$, Poisson-commuting, globally defined functions $I_1, \ldots, I_N: M\to \mathbb{R}$, so that the Hamiltonian $H$ is a function of $I_1, \ldots, I_N$, only.
Such integrable system is called maximally superintegrable if
there additionally exist $N-1$ globally defined integrals of motion $I_{N+1}, \ldots, I_{2N-1}: M\to \mathbb{R}$, so that the combined set $(I_{1}, \ldots, I_{2N-1})$ is functionally independent.
It follows from Caratheodory-Jacobi-Lie theorem that every finite-dimensional autonomous Hamiltonian system on a symplectic manifold $(M,\omega)$ is locally maximally superintegrable in sufficiently small local neighborhoods around any point of $M$ (apart from critical points of the Hamiltonian).
The main point is that (global) integrability is rare, while local integrability is generic.
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$^1$ An autonomous Hamiltonian system means that neither the Hamiltonian $H$ nor the symplectic two-form $\omega$ depend explicitly on time $t$.
$^2$ Outside differential geometry $N$ functions $I_1, \ldots, I_N$ are called functionally independent if $$\forall F:~~ \left[z\mapsto F(I_1(z), \ldots, I_N(z)) \text{ is the zero-function} \right]~~\Rightarrow~~ F \text{ is the zero-function}.$$
However within differential geometry, which is the conventional framework for dynamical systems, $N$ functions $I_1, \ldots, I_N$ are called functionally independent if $\mathrm{d}I_1\wedge \ldots\wedge \mathrm{d}I_N\neq 0$ is nowhere vanishing. Equivalently, the rectangular matrix $$\left(\frac{\partial I_k}{\partial z^K}\right)_{1\leq k\leq N, 1\leq K\leq 2N}$$ has maximal rank in all points $z$. If only $\mathrm{d}I_1\wedge \ldots\wedge \mathrm{d}I_N\neq 0$ holds a.e., then one should strictly speaking strip the symplectic manifold $M$ of these singular orbits.
Best Answer
As you point out, there's an issue with the notion of quantum integrability defined, in analogy with the classical case, through the existence of sufficiently many conserved charges. For more on this, see Caux and Mossel, "Remarks on the notion of quantum integrability" https://arxiv.org/abs/1012.3587
So instead one can ask for a underlying algebraic notion, called 'Yang--Baxter integrability', that is well defined and implies the existence of many conserved charges. For more about this, see also https://physics.stackexchange.com/a/780318/.
The commuting charges constructed in this way (as logarithmic derivatives of the transfer matrix) are less and less local, see How local are the conserved charges in a quantum integrable model?. However, one can construct so-called quasi-local charges as the (first) logarithmic derivative of transfer matrices with higher spin in the auxiliary space (rather than higher derivatives of the 'fundamental' transfer matrix with spin-1/2 auxiliary space). See Ilievski, Medenjak and Prosen, "Quasilocal conserved operators in isotropic Heisenberg spin 1/2 chain" (https://arxiv.org/abs/1506.05049) and Ilievski et al, "Complete Generalized Gibbs Ensemble in an interacting Theory" (https://arxiv.org/abs/1507.02993).