Finding all solutions to a system of non-linear equations is a longstanding and challenging problem with applications across science and engineering.  The recent emergence of large language models raises the question of whether these models can assist, or even autonomously solve, such problems. This paper reviews classical and global optimization approaches for finding all roots to systems of nonlinear equations and empirically evaluates nine contemporary large language models on a suite of small-sized representative systems. The solution quality is analyzed, common failure modes are investigated, and the impact of prompt design is considered, offering guidelines for practitioners and directions for future research.