Let $\mathbb{S}^2 \subseteq \mathbb{R}^3$ be the unit sphere. Let $\gamma : [0, l] \to \mathbb{S}^2$ be a smooth embedded closed curve. Let $\vec{\nu}$ be one of the two continuous unit normal vector field on $\gamma$ (with a little abuse, I am denoting with $\gamma$ also the image of the curve). Just to be clear, $\vec{\nu}$ is tangent to the sphere.
Let $f: \gamma \to \mathbb{R}$ be a smooth, given function, with only a finite number of zeroes. Let assume also that the following property holds: for every $K \in \mathcal{Kill}(\mathbb{S}^2)$ \begin{equation}\label{asd} \int_\gamma f \langle \nu, K\rangle \, d\sigma = 0. \end{equation}
With $\mathcal{Kill}(\mathbb{S}^2)$ I mean the Lie algebra of Killing vector fields on the sphere.
Question: what can I say about $\gamma$ and/or $f$?
For instance, it should be true that if $\gamma$ is a circle and $f$ is constant, then $\int_\gamma f \langle \nu, K\rangle \, d\sigma = 0$ holds for every $K \in \mathcal{Kill}(\mathbb{S}^2)$. Is it true the converse? I feel that the converse might not be true, but maybe it is possible to prove some weaker kind of symmetry for $\gamma$ and/or $f$?