Let us consider a function $f: [a,b]\times S^3 \rightarrow \mathbb{R}^2$ where $a,b$ are reals and $S^3$ is the sphere defined by $x^2+y^2+z^2+w^2=1$.
I will denote $\vec{u} = (k,\vec{r})$ where $k\in [a,b]$ and $\vec{r} = (x,y,z,w)\in S^3$. Then, the function $f$ has two-components
$$ \vec{f}(\vec{u}) = \left( f_1(\vec{u}),f_2(\vec{u})\right) $$
In am interested in the following (very general) problem.
Find if it exists a set of positive coefficients $\{a_k>0\}$ such that
$$ \sum_n a_n\, \vec{f}(\vec{u}_n) =0. $$
For $n>2$ this is equivalent of asking if I can find a closed path built with "vectors" proportionals to $\vec{f}(\vec{u}_n)$.
Are these kind of problems studied somewhere?
Take for example the functions
$f_1(\vec{u}) = 9 (144 - 528 k + 532 k^2 - 200 k^3 + 25 k^4) w^2 - 60 x^2 + 10 (8 - 12 k + 3 k^2) y^2 + 900 z^2 - 1872 k z^2 + 1188 k^2 z^2 - 360 k^3 z^2 + 45 k^4 z^2\\ f_2(\vec{u}) = 12 (17 - 24 k + 6 k^2) w^2 - 15 x^2 + 5 (8 - 12 k + 3 k^2) y^2 + 9 (17 - 24 k + 6 k^2) z^2$