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We ask two related questions which are inspired by this MO question Does $P_xP_y+Q_xQ_y=0 \implies$ "NONEXISTENCE OF LIMIT CYCLE for $P\partial_x+Q\partial_y$"? (Complex Dilatation and Limit cycle Theory)

1)Let $M$ be a parallelizable manifold and $X$ be a (non vanishing) vector field on $M$. Is there a Riemannian metric $g$ on $M$, with associated Sasakian metric $g_S$ on $TM$, such that $DX:TM\to TTM$ carries a golabal orthonormal frame $V_1,V_2,\ldots,V_n$ to a mutually orthogonal set?

2)Let $(M,g)$ be a parallelizable Riemannian manifold with associated Sasakian metric $g_S$ on $TM$. Assume that $X$ is a (non vanishing) vector field on $M$. Is there a golabal orthonormal frame $W=\{V_1,V_2,\ldots,V_n\}$, such that $DX:TM\to TTM$ carries $W$ to a mutually orthogonal set?

What kind of dynamical obstructions would appear?

Remark: As a relation between the two questions of this post and its inspiring post Does $P_xP_y+Q_xQ_y=0 \implies$ "non-existence of limit cycle" for $P\partial_x+Q\partial_y$"? (Complex dilatation and limit cycle theory) we notice that every vector field $X=P\partial_x+Q\partial_y$ which satisfies $P_xP_y+Q_xQ_y=0$ then its derivative $DX:TR^2 \to TT \mathbb{R}^2 $ sends the standard orthonormal frame of $\mathbb{R}^2$ to a set of orthogonal vectores.

Note: By parenthesized word (non vanishing) mentioned in the above two questions, we would like to say:Our preference is a non vanishing vector field but vanishing examples are also very appreciated.

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