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The purpose of this work is to explore possible side conditions involving high order differential invariants with the aim of reducing the original mathematical model. It is shown that, in the case of the Tzitzeica curve equation, a suitable side condition that leads to exact solutions is a system consisting of two third order differential invariants involving the arbitrary functions and a third order differential invariant involving the dependent variable of the equation. In this situation, the equation can be reduced to a linear equation for the equation's constant. Similarly, in the case of a proposed generalization of the Tzitzeica curve equation, the above side condition also leads to a reduced model.