This paper studies the Mei symmetry and Mei conserved quantity of non-Chetaev's type in the event space. The differential equations of motion of non-holonomic s ystems of non-Chetaev's type in event spaces are established. The definition and the criteria of Mei symmetry are given. The condition and the form of Mei conse rved quantity are deduced directly from the Mei symmetry. An example is given to illustrate the application of the results.

Hojman conserved quantities deduced by using the special Lie symmetry, the Noether symmetry and the Mei symmetry for systems with non-Chetaev nonholonomic constraints in the event space are studied. First, the differential equations of motion for the above systems are established. Second, the criterion of the Lie symmetry, the Noether symmetry, the Mei symmetry and the relation between them are obtained. Third, the conservation law obtained by Hojman is generalized and applied to the systems, and Hojman conserved quantities are obtained. An example is given to illustrate the application of the results.

Lie symmetry of Appell equation and conserved quantity deduced directly by Lie symmetry for a Chetaev's type constrained mechanical system are investigated. The relations between Lagrange function and A function are analyzed. A general approach of studying conserved quantity deduced by Lie symmetry of Appell equation for a Chetaev's type constrained mechanical system is discussed. The definition and the criterion of Lie symmetry of Appell equations under the infinitesimal transformations of groups are given. The structural equation of Lie symmetry and the expression of conserved quantity deduced directly by Lie symmetry are obtained. An example is given to illustrate the application of the results.

The Mei symmetry of Tzénoff equations for nonholonomic systems under the infinitesimal transformation of groups is studied in this paper. The definition and the criterion equations of the symmetry are given. If the symmetry is a Lie symmetry, then the Hojman conserved quantity of the Tzénoff equation can be obtained by the Mei symmetry.

Mei symmetry and Mei conserved quantity of Nielsen equation with multipliers for a nonholonomic， non-conservative system of Chetaev's type are studied- The differential equations of motion of Nielsen equation with multipliers for the system，the definition and criterion of Mei symmetry，and the condition and the form of Mei conserved quantity deduced directly by Mei symmetry for the system are obtained- An example is given to illustrate the application of the results-