Integrability of Equations Admitting the Nonsolvable Symmetry Algebra so(3,R)

If an ordinary differential equation admits the nonsolvable Lie algebra     , and we use any of its generators to reduce the order, the reduced equation does not inherit the remaining symmetries. We prove here how the lost symmetries can be recovered as   C ^∞  -symmetries of the reduced equation. If the order of the last reduced equation is higher than one, these   C ^∞  -symmetries can be used to obtain new order reductions. As a consequence, a classification of the third-order equations that admit     as symmetry algebra is given and a step-by-step method to solve the equations is presented.