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Let $M$ be a smooth manifold of (real) dimension $2n$. An almost complex structure $J$ on $M$ is a linear vector bundle isomorphism $J \colon TM\to TM$ on the tangent bundle $TM$ such that $J^2 = − 1 \cdot id_{TM}$.

Intuitivlely, that's a continuous fibrewise "choice" of $J_p \in GL(T_pM)$ turning $T_pM $ into complex vspace with complex unit $i_p:= J_p$. On level of vector bundle theory the existence of an almost complex structure on manifold $M$ is equivalent to reduction of the structure group of tangent bundle $GL(2n, \Bbb R)$ to $GL(n, \Bbb C)$, so a statement purely in terms of algebraic topology.

Clearly a complex structure on $M$ canonically induce on level of induced transition maps almost complex structure, but the converse is in general a complicated issue in real dimension $n >1$ in general wrong.

Question: I read several times that when a.c. structure $J$ admits extension to honest complex structure on $M$, one often calls "$J$ to be "integrable". "Integrable" in which sense?
If course one could just "absorb" this term as definition and turn on to work with it, but I wonder if there is a figurative reason/motivation for choosing that term "integrable" to describe liftability of $J$.

Could somebody elaborate what is the background of the term "integrable" and its geometric intuition one should keeping in mind thinking about it in this context adressing the "liftability" of almost complex structures to complex?

Is there a historically relevant example(s) having actually something to do with "naive integration" or/and motivating the term "integrable" here; or is this more related with so called "integrable systems". (The latter I conjectured firstly, that maybe the liftability of $J$ can be somehow boiled down to integrability in latter sense of some appropriete system of differential equations, but that's just a guess, I don't actually haven't found something in that direction, that's why I'm asking for.

Eg, presumably the most prominent criteria when $J$ actually admits extension to complex - the Newlander–Nirenberg theorem - phrased in terms of vanishing of certain tensor $N_J$ - does not make use of such hypothetical integrability condition for certain DGLs.

#EDIT (added later): Let me add this answer from MO, so "integrability" here seems as Gabe K in comment below already indicated indeed to come from "Frobenius integrability".

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    $\begingroup$ In the Newlander-Nirenberg theorem, the term “integrable” is used in the same way as for the Frobenius theorem for when a collection of vector fields induces a foliation of a manifold. Indeed, for smooth enough ACS, the Newlander-Nirenberg theorem can be proven directly by the Frobenius theorem. However, for less smooth structures it is very nontrivial. $\endgroup$
    – Gabe K
    Commented Dec 6 at 17:55

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A general type of question is if you have something that looks like a differential of something else, is it really the differential of something. If there is, we call the system integrable. If the manifold has dimension greater than $1$, there is always a necessary condition arising from the fact that partials commute. This is called an integrability condition.

An almost complex structure is a complex structure if there exist local complex coordinate functions where the almost complex structure can be written in terms of the differentials of coordinate functions. That partials commute is in this setting equivalent to the Newlander-Nirenberg equations. If the equations hold, the almost complex structure can be “integrated” to obtain the coordinate functions.

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