$\mathbf {The \ Problem \ is}:$ Let $S$ be the set of all lines through $(0,y)$ parallel to $x-$axis where $y \in \mathbb R-\mathbb Q$ . Is it an integral submanifold for $V=\frac{\partial}{\partial x}?$
$\mathbf {My \ approach}:$ Take $p=(0,√2)\in S$ ,the integral curve of $V$ via $p$ is $\{(t,√2)| t\in \mathbb R\}.$
S is disconnected $1$-dimensional manifold .
Now, I can't see why $S$ can't be an integral submanifold of $V$ so I think $S$ will be .
A small hint is required, thanks in advance .