The aim of this paper is to study a particular reduction case of the nonlinear ordinary differential equation that defines a Tzitzeica curve. It is shown that the Tzitzeica curve equation can be reduced to an auxiliary third order linear homogeneous ordinary differential equation with constant coeficients for the defining functions of the curve and a linear equation for the equation's constant. Consequently, it can be proven that any three linearly independent solutions of the auxiliary ordinary differential equation define a Tzitzeica curve.
Bila, Nicoleta V. and Eni, Michael, "Particular solutions to the Tzitzeica curve equation" (2012). Math and Computer Science Working Papers. Paper 15.