Eigenfunctions of distinct eigenvalues with the same winding number are never parallel

Let $S_\infty:\mathbb{R}\to\mathbb{R}^{2\times2}$ be a $C^\infty$ $1$-periodic loop of symmetric matrices and $A=-J_0\partial_t-S_\infty$ on $1$-periodic loops in $\mathbb{R}^2$. Let $\mu_1\ne\mu_2$ be eigenvalues of $A$ such that every eigenfunction of $\mu_1$ and every eigenfunction of $\mu_2$ has the same winding number $k\in\mathbb{Z}$. If $e_1,e_2$ are eigenfunctions with eigenvalues $\mu_1,\mu_2$, then for every $t\in\mathbb{R}$ and every $c\in\mathbb{R}$,

The values of such eigenfunctions are linearly independent in $\mathbb{R}^2$ at every time. This controls how eigenfunctions of one winding number can combine.

Formalization Note Eigenfunctions are IsLoopEigen Sinf μ e (FiniteEnergyPlane_Asymptotics); the winding hypothesis on $\mu_j$ is HasLoopEigenWind Sinf μⱼ k (FiniteEnergyPlane_Winding: $\mu_j$ has an eigenfunction and all its eigenfunctions have winding $k$ as loops in $\mathbb{C}$ via vecToC); IsSymLoop Sinf and smoothness of the entries are hypotheses.