indeed symmetric.

Next, we prove a Birman-Schwinger principle for the operator $A_{\eta, \tau}$. This relates the linear eigenvalue problem for the differential operator $A_{\eta, \tau}$ to the nonlinear eigenvalue problem for a family of bounded integral operators involving the maps $\mathcal{C}_\lambda$ introduced in , which yields also a reduction of the space dimension for the eigenvalue problem. We would like to note that this lemma can only be shown in this simple form for noncritical interaction strengths, i.e. for $\eta^2 - \tau^2 \neq 4 c^2$. The result stated below follows from the general consideration in [@BEHL17 Theorem 2.4] or [@AMV15 Proposition 3.1], but to keep the paper self-contained, we add the short simple proof here.

\[lemma\_Birman\_Schwinger\] Let $\eta, \tau \in \mathbb{R}$ such that $\eta^2 - \tau^2 \neq 4 c^2$ and let the operator $A_{\eta, \tau}$ be defined by .

-   If for $\lambda \in \rho(A_0)$ and $\varphi \in L^2(\Sigma)^4$ one has $\big(I_4 + (\eta I_4 + \tau \beta) \mathcal{C}_\lambda \big) \varphi \in H^{1/2}(\Sigma)^4$, then it follows $\varphi \in H^{1/2}(\Sigma)^4$.

-   $\lambda \in \rho(A_0) \cap \sigma_{\textup{p}}(A_{\eta, \tau})$ if and only if $-1 \in \sigma_\textup{p}\big((\eta I_4 + \tau \beta) \mathcal{C}_\lambda \big)$.

-   For $\lambda \in \mathbb{C} \setminus \mathbb{R}$ the inverse $$\big( I_4 