One example of a nonholonomic constraint is a disk rolling around in the cartesian plane that is constrained to not be slipping.
These leads to the constraint $dx - a \sin\theta d\phi = 0$ and $dy - a\cos\theta d\phi = 0$ Where $\phi$ is the angle of how far the disk has rotated, $-\theta$ is angle that velocity makes with respect to $x$.
We know that these aren't exact differentials because to put it in form $Mdx + Ndy = 0$, we don't have $\partial M/\partial y = \partial N / \partial x$. But that doesn't mean we can't find some integrating factor to multiply it by and make it an exact differential? So I don't see how we know that is it nonintegrable?
If I take $f(x,\phi)[ dx - a \sin\theta d\phi] = 0$ and try to manipulate it to get something analogous to $\partial M/\partial y = \partial N / \partial x$, then I get a tricky PDE, which I don't know how to solve.