An **arc field** on a topological space $X$ is a continuous function $\Psi: X \rightarrow X^{[0,1]}$ such that for every $x \in X$, the path $\Psi(x): [0,1] \rightarrow X$

(1) starts at $x$,

(2) is either an embedding (in other words a simple path), or the constant path on $x$. In the latter case we call $x$ a **singularity** of $\Psi$. Here $X^{[0,1]}$ is equipped with the compact-open topology.

This kind of definition that generalizes the notion of a vector field (or/and the tangent bundle) from smooth manifolds to the topological setting, goes back at least to Nash (MR0071081, A path space and the Stiefel-Whitney classes, 1955). There are results by Brown, Fadell, Stern (MR0173269, MR0328952, MR0310880) which show that certain versions/corollaries of Hopf's index theorem generalize using arc fields. For example the following is true:

Suppose $X$ is either a compact topological manifold of dimension $\neq 4$, or a finite simplicial complex such that every punctured neighborhood $U - \{x\}$ is connected provided that $U$ is connected. If $X$ has zero Euler characteristic, it admits an arc field with no singularities.

My question is about moving the singularities around continuously when they are unavoidable. Feel free to assume $X$ is a compact smooth(able) manifold with nonzero Euler characteristic, but the more general the better of course. Write $\text{PConf}_{k}(X) := \{(x_{1}, \dots, x_{k}): x_{i} \neq x_{j}\}$ for the $k$-fold ordered configuration space of $X$. Does there exist a finite $k$ and a continuous map $$\Theta: \text{PConf}_{k}(X) \times X \rightarrow X^{[0,1]}$$ such that for every configuration $(x_{1}, \dots ,x_{k})$, the map $\Theta(x_{1}, \dots, x_{k}, - ): X \rightarrow X^{[0,1]}$ is an arc field whose singularities are contained in $\{x_{1}, \dots, x_{k}\}$?

I don't know the answer even for the sphere $S^2$, but have a convoluted way of showing that $k = 1$ won't do it. It feels like $k = 2$ should be enough for $S^2$, but I don't really know.