This paper presents an efficient algorithm called SolveFuzzySystem, or SFS, for finding real solutions of polynomial systems whose coefficients are symmetrical L-R fuzzy numbers with bounded support for which the spread functions L and R are bijective. The real solutions of such a system are deduced from solutions of some polynomial systems with real coefficients. This algorithm is based on new results that are universal because they are independent from the spread functions. These theoretical results include the management of the fuzzy system's solutions signs. An implementation in the Fuzzy package of the free computer algebra software SageMath and a parallel version of the algorithm are described.