This paper investigates the nonlinear Maxwell–Dirac system, which describes the time-evolution of relativistic electrons and positrons within external and self-consistent electromagnetic fields. The system is given by a set of partial differential equations involving the wave function \(\psi\), the magnetic field \(\mathbf{A}\), the electron field \(\phi\), and the charge distribution \(q\). The authors focus on the case where the potential \(V\) has a local trapping potential well, and they construct an infinite sequence of localized bound state solutions that concentrate around the local minimum points of \(V\). These solutions are of higher topological type, obtained from a symmetric linking structure. In the second part, the authors consider the degenerate case where \(V(x)\) approaches \(a\) as \(|x| \to \infty\), which is a more general scenario compared to the strict gap condition typically assumed in previous studies. The paper provides a detailed analysis of these solutions and their properties, contributing to the understanding of the nonlinear Maxwell–Dirac system in the presence of degenerate potentials.This paper investigates the nonlinear Maxwell–Dirac system, which describes the time-evolution of relativistic electrons and positrons within external and self-consistent electromagnetic fields. The system is given by a set of partial differential equations involving the wave function \(\psi\), the magnetic field \(\mathbf{A}\), the electron field \(\phi\), and the charge distribution \(q\). The authors focus on the case where the potential \(V\) has a local trapping potential well, and they construct an infinite sequence of localized bound state solutions that concentrate around the local minimum points of \(V\). These solutions are of higher topological type, obtained from a symmetric linking structure. In the second part, the authors consider the degenerate case where \(V(x)\) approaches \(a\) as \(|x| \to \infty\), which is a more general scenario compared to the strict gap condition typically assumed in previous studies. The paper provides a detailed analysis of these solutions and their properties, contributing to the understanding of the nonlinear Maxwell–Dirac system in the presence of degenerate potentials.