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The solve_mod command might be useful:

Return all solutions to an equation or list of equations modulo the given integer modulus. Each equation must involve only polynomials in 1 or many variables.

Type solve_mod? for the help message and examples. This is most useful when there are relatively few solutions, maybe even when there are relatively few possible solutions. It is very slow for the particular equations you provide, partly because the modulus is 31 and it may very well be trying all possible values for x, y, and z. Also, it will list all solutions, not solve it symbolically the way you are asking.