As cyclic codes and maximum distance separable (MDS) codes, cyclic MDS codes have very nice structures and properties, which have been intensively investigated in literature due to their theoretical interest and practical importance. Particularly, abundant cyclic MDS codes have been determined and constructed for many parameters and most of them were proved to be equivalent to generalized Reed-Solomon (GRS) codes. Hence it is a challenging task to construct non-Reed-Solomon type cyclic MDS codes. In this work, we obtain many new cyclic MDS codes for certain parameters by determining the solutions of the system of polynomial equations. Moreover, by determining the dimension of the Schur square of an MDS code, we can easily show that all of our constructed codes are not equivalent to GRS codes.