Inspired by the following two posts
Finding a 1-form adapted to a smooth flow
whose subjects are potentially related to limit cycle theory, we ask the question below:
Question: Is there a vector field $X=P\partial_x +Q\partial_y$ on $\mathbb{R}^2$ which satisfy $P_xP_y+Q_xQ_y=0$ but $X$ is not a geodesible vector field on $X\setminus S$ where $S$ is the set of singular points of $X$.
Remark 1: A geodesible vector field is a vector field $X$ which admit a Riemannian metric $g$ on $X\setminus S$, the complement of its singular set, such that all non singular solution curves of $X$ are unparametrized geodesics with respect to metric $g$.
Remark 2: There are two class of vector fields which lies in both class "gedesible vector fields" and "vector fields $X$ whose two columns of Jacobian matrix $JX$ are perpendicular to each other: The holomorphic one and $f(x)\partial_x+g(y)\partial_y$
