$M$ compact manifold. Let $\lambda$ be an eigenvalue for the Dirac operator of multiplicity greater than 2. I'm interested in showing the existence of two linearly independant eigenspinors $u$ and $v$ of eigenvalue $\lambda$, with $\|u\|_2=\|v\|_2=1$ and so that there exists a point $x_0 \in M$ for which $\|u(x_0)\|_{x_0} \neq \|v(x_0)\|_{x_0}$. Trying to show this by contradiction, I assumed that $\|u(x)\|_x = \|v(x)\|_x$ for all $x \in M$. I can prove the desired result provided there be two linearly independant eigenspinors of eigenvalue $\lambda$ with $\langle u,v \rangle_2 \neq \frac{\langle u(x),v(x) \rangle_x}{|u(x)|_x^2}$ for some $x \in M$.

My question is:

Is it possible that any two normalized linearly independant eigenspinors $u$,$v$ in such $\lambda$ eigenspace satisfy both $$\|u(x)\|_x = \|v(x)\|_x \quad \quad \text{and} \quad \quad\langle u,v \rangle_2 =\frac{\langle u(x),v(x) \rangle_x}{|u(x)|_x^2} \quad \text{for all } x \in M \;?$$