If $X$ is a Hausdorff space, by a flow I mean a continuous surjection $F: X \times \mathbb{R} \rightarrow X$ such that $F(x, s + t) = F(F(x,s), t)$ for all $x \in X$ and $s, t \in \mathbb{R}$. If $x$ is periodic (or fixed) under $F$, then let $p_F(x)$ denote the minimal non-negative time such that $F(x, p_F(x)) = x$. Otherwise, write $p_F(x) = \infty$.
In other words, $p_F(x)$ is the length of the period of $x$, and is infinite when $x$ is not periodic.
Then by a continuity argument you can show that the function $p_F(x)$ is lower semi-continuous from $X$ into $[0, \infty]$; especially, if $x_n \rightarrow x$ and $ 0 < p_F(x) = r < \infty$, then every finite accumulation point of $\lbrace p_F(x_n) \rbrace$ is a positive integer multiple of $r$, as necessary due to the continuity of F.
What lsc functions $f: \mathbb{R}^2 \rightarrow \mathbb{R}$ arise as $p_F$ for some plane flow $F$?
It's not all of them: For example, if $f$ is constantly $1$ except at a single point, where it takes the value $\frac{1}{2}$. By the above argument, any jump discontinuity must be an integer multiple of the lower limit value, but the previous example shows that this is not sufficient. Note that in the plane any non-periodic orbit of a flow is homeomorphic to $\mathbb{R}$, providing another 'obvious' constraint.
What are some other properties that might be sufficient to allow an associated flow?