A sixth-order Sturm-Liouville equation of the form \[ Ly:=-y^{(6)}+(A(z)y'')''+(B(z)y')'+C(z)y=\lambda y, \;\;\;\;a\leq z\leq b, \] is investigated with six end conditions which make a self adjoint problem. The Sturm Liouville operator is factorized as the product of a third order differential operator and its adjoint. It is shown that factorization of the operator leads to a system of nonlinear third-order ordinary differential equations, the so-called principal system. The principle system is solved by using Lie symmetry methods and it is shown that it may admit a one or two parameter Lie group of transformations. Using this fact, in case 1 a class of isospectral operators is obtained. In case 2, the principle system is transformed to Chazy's equation which admits a three parameters group of transformations. In case 3, the sixth-order isospectral operators is produced.