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Assume that $$\begin{cases}\dot x=P(x,y)\\\dot y=Q(x,y)\end{cases}$$ is a non vanishing holomorphic vector field on an open subset $U$ of $\mathbb{C}^{2}\simeq \mathbb{R}^{4}$. It defines a two dimensional real distributation on $U$. Of course it is an integrable real distribution.

Now consider the real $2$ dimensional distribution on $U$ which is orthogonal to the above distribution.(With respect to the heremitien inner product of $\mathbb{C}^{2}$).

Is the later distribution integrable? Can the leaves be parametrized by holomorphic functions in $t\in \mathbb{C}$?

To what extend this orhogonal distributions are studied for singular holomorphic foliation of $\mathbb{C}P^{2}$?

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    $\begingroup$ It's not hard to show that this can only happen if the ratio $[P:Q]$ is locally constant. The point is that the orthogonal foliation is annihilated by the real and imaginary parts of the (1,0)-form $\alpha = \overline{P}\,\mathrm{d}x + \overline{Q}\,\mathrm{d}y$, and the condition for integrability of the complex line field is the vanishing of $\alpha\wedge\overline{\alpha}\wedge\mathrm{d}\alpha$, which only vanishes identically when the ratio of $P$ to $Q$ is locally constant. $\endgroup$
    – Robert Bryant
    Commented Nov 17, 2015 at 12:35
  • $\begingroup$ @RobertBryant Thank you for your answer. Can one imagine a possible relation between completely nonintegrability of the later distribution and the dynamics of the initial foliation(vector field)? $\endgroup$
    – Ali Taghavi
    Commented Nov 17, 2015 at 16:24

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