Symmetries of First Integrals and Their Associated Differential Equations

The relationship between the reduction of order through point symmetries and integration is explored with particular emphasis on the loss and gain of point contact for third order.symmetries to and from nonlocal symmetries. It is seen that reduction of order can even lead to the loss of all point symmetries at the third order level and their replacement at the second order level from nonlocal symmetries. It is evident that nonlocal symmetries should be given more attention in applications.