Tali — Multiplying a (smooth) vector field by a positive (smooth) function is only one of several ways to possibly obtain another vector field with equivalent orbits. For instance, for any "structurally stable" vector field V in the sense of Smale after undergoing a sufficiently small C¹ perturbation will result in a field V' that is orbit equivalent to V. But be aware that the equivalence may be only by a homeomorphism, and not a diffeomorphism, of the underlying manifold that is close to the identity. Another way is to carry V to a new one V' by any diffeomorphism of the manifold.

Hurwitz's theorem states that for a Riemann surface of genus 2 or more, the size of its full automorphism group is no greater than 84(g-1). Any non-trivial holomorphic flow consists of infinitely many automorphisms, so cannot exist on a higher genus Riemann surface.

On a meta level, I find it tremendously unfortunate that a perfectly valid question like this one, by someone who perhaps has not studied this area very much, needs to be closed off. It smacks of snobbery and elitism in the worst way. I cannot imagine how allowing such questions does any harm to MO in any way.

If the vector field is "complete" (meaning its solution curves and hence its flow) exist for all complex times t), then for each time t the flow at that time is a biholomorphism.

I suspect much confusion stems originally from George Gamow's book "One Two Three ... Infinity" — which got a number of things wrong about the continuum and the continuum hypothesis, which it implied was settled. None of the reprintings of this book fixed the error — to this day.

The orientation-preserving isometries of the circle is a circle. In general the identity component of the isometry group of an n-torus R^n / L (where L is an n-dimensional lattice in R^n) is the very same n-torus acting on itself by addition. But it would still be nice to find a Riemannian structure on a Lie group G whose full isometry group is the same Lie group G.

The surface of genus 2 has a diffeomorphism of order 5: Imagine a "carousel" of two parallel closed decagonal 2-disks in 3-space that are connected by 5 equally spaced half-twisted strips between their alternate corresponding edges. This surface has just one boundary component, which we cap off with a 2-disk. It's then easy to check that this is an orientable surface of genus 2, and that rotating the carousel by 2π/5 extends to the 2-disk cap, thereby effecting a diffeomorphism of order 5.