In this brief, linear quadratic infinite-horizon Nash games for general multiparameter singularly perturbed systems are studied. The local uniqueness and the asymptotic structure of the solutions to the cross-coupled multiparameter algebraic Riccati equation (CMARE) are newly established. Utilizing the asymptotic structure of the solutions to the CMARE, the parameter-independent Nash strategy is established. A numerical example is given to demonstrate the efficiency and feasibility of the proposed analysis.Index Terms-Cross-coupled algebraic Riccati equation (CARE), general multiparameter singularly perturbed systems (GMSPS), local uniqueness, Nash games, parameter-independent Nash strategy.