The multiple solution problem of nonlinear energy sinks (NESs) is of great interest in academic research and engineering applications; however, there exists only limited work on the multiple solution branches of piecewise linear NESs. Piecewise linear NESs can give rise to the appearance of isolated solution branches (ISBs), which are harmful to the performance of dynamical systems. It is difficult to detect the ISBs of a dynamical system with a piecewise linear NES. Hence, in this study, the multiple periodic solution branch distribution of a four-degree-of-freedom dynamical system with a piecewise linear NES is investigated. A method based on polynomial homotopy that can capture ISBs is developed to search multiple solution branches. Spurious periodic solutions can exist owing to the use of the implicit polynomial approach; therefore, the harmonic balance method with high harmonic orders and alternating frequency/time-domain technique are employed to distinguish true periodic solutions. Numerical experiments demonstrate that the proposed method can search for multiple periodic solution branches. A parameter analysis is conducted on the multiple solution branches, and the inner and outer ISBs are observed.