Exercises from section 14.3 of Katok's Introduction to the Modern Theory of Dynamical Systems

Question

I am reading the book "Introduction to the Modern Theory of Dynamical Systems" by Anatole Katok and Boris Hasselblatt. I find myself in section 14.3 which deals with minimal sets (specifically from Schwartz's theorem). I am trying to do the following exercises:

1. Show that any $C^1$ flow on the orientable surface of genus $g$ has no more than $g$ different minimal sets that are not fixed points or periodic orbits.
2. Given an orientable surface of genus $g$ and $1\leq k\leq g$ show that there exists a $C^1$ flow with exactly $k$ nowhere-dense minimal sets that are not fixed point or circles.

I don't know how to attack the problems, for example if $M$ were an orientable manifold of genus $3$ then it can be thought of as a $3-$torus then using Schwartz's theorem comes to mind:

Let $M$ be a two-dimensional differentiable manifold of class $C^2$ compact and connected. Let $\varphi: \mathbb R \times M \to M$ be a flow of class $C^2$ in $M$. A minimal set $\mu\subset M$ can be either:

1. a singular point, or
2. a periodic orbit, or
3. an entire manifold $M$ which is homeomorphic to the torus $\mathbb T^2$.

but I can't really see how this can help me since the flow is of class $C^1$. Any ideas please?

I found the following hint given by the author, for the first question:

Use the fact that any $g+1$ disjoint closed curves divide the surface, and the Poincaré-Bendixson theorem.