Suppose $(X,I)$ is an almost-complex real analytic manifold where $I$ is a real-analytic almost complex structure. Suppose there exists an $I$-almost complex submanifold $M\subset X$ where this means that $M$ is a real analytic submanifold such that given a tangent vector $v$ to $M,$ then $I(v)$ is also tangent to $M.$ Suppose that the restriction of $I$ to $M$ is integrable. The question is: are there conditions under which this implies that $I$ is integrable on all of $X?$ I would be supremely surprised if the condition was empty, namely that this always implied that $I$ was integrable, although I wouldn't look a gift horse in the mouth. Nevertheless, a condition such as the restriction of $I$ to $M$ being Stein, or something of such a global nature, doesn't sound quite so ridiculous.
I should add that in the exact situation I'm considering, $X$ is non-compact (it's topologically a ball) and $M$ is non-compact (also topologically a ball) and the restriction of $I$ to $M$ is Stein, and lots of other nice things you might dream it to be as a complex manifold.
Edit: I've just realized that I need to add that $M$ is not of complex dimension one, otherwise the integrability of $I$ restricted to $M$ is a trivial consequence of there being no $(2,0)$-forms on $M.$
Thanks for any comments or ideas you might have.