  The solution of a class of third order ordinary differential equations
possessing two parameter Lie symmetry group is obtained by group theoretic
means. It is shown that reduction to quadratures is possible according to two
scenarios: 1) if upon first reduction of order the obtained second order
ordinary differential equation besides the inherited point symmetry acquires at
least one more new point symmetry (possibly a hidden symmetry of Type II). 2)
First, reduction paths of the fourth order differential equations with four
parameter symmetry group leading to the first order equation possessing one
known (inherited) symmetry are constructed. Then, reduction paths along which a
third order equation possessing two-parameter symmetry group appears are
singled out and followed until a first order equation possessing one known
(inherited) symmetry are obtained. The method uses conditions for preservation,
disappearance and reappearance of point symmetries.
