This paper studies a nonlinear Maxwell–Dirac system, which describes the time-evolution of relativistic electrons and positrons in electromagnetic fields. The system is given by a set of equations involving the Dirac matrices, the electromagnetic field, and the electron field. The paper focuses on constructing an infinite sequence of localized bound state solutions for this system under certain conditions. These solutions concentrate around local minima of a potential function V, which is assumed to have a local trapping potential well. The solutions are of higher topological type, obtained from a symmetric linking structure. In the second part of the paper, the potential V is considered to approach a certain value as |x| → ∞, which is a degenerate case compared to the usual strict gap condition. This paper presents a new approach to handle the degenerate potential case, which is important for the variational formulation and compactness issues. The study shows that an unbounded sequence of bound states can be constructed under these conditions. The paper also discusses the mathematical structure of the system, including the definitions of the Dirac matrices and the electromagnetic field. The stationary wave solutions are defined in terms of a wave function w(x) and the electromagnetic fields A and A₀. The paper provides a detailed analysis of the system's mathematical structure and the conditions under which the bound state solutions can be constructed. The results contribute to the understanding of the nonlinear Maxwell–Dirac system and its potential applications in physics.This paper studies a nonlinear Maxwell–Dirac system, which describes the time-evolution of relativistic electrons and positrons in electromagnetic fields. The system is given by a set of equations involving the Dirac matrices, the electromagnetic field, and the electron field. The paper focuses on constructing an infinite sequence of localized bound state solutions for this system under certain conditions. These solutions concentrate around local minima of a potential function V, which is assumed to have a local trapping potential well. The solutions are of higher topological type, obtained from a symmetric linking structure. In the second part of the paper, the potential V is considered to approach a certain value as |x| → ∞, which is a degenerate case compared to the usual strict gap condition. This paper presents a new approach to handle the degenerate potential case, which is important for the variational formulation and compactness issues. The study shows that an unbounded sequence of bound states can be constructed under these conditions. The paper also discusses the mathematical structure of the system, including the definitions of the Dirac matrices and the electromagnetic field. The stationary wave solutions are defined in terms of a wave function w(x) and the electromagnetic fields A and A₀. The paper provides a detailed analysis of the system's mathematical structure and the conditions under which the bound state solutions can be constructed. The results contribute to the understanding of the nonlinear Maxwell–Dirac system and its potential applications in physics.