Symmetries and first integrals of a second order evolutionary operator equation

The author considers the evolutionary operator equation N(u)equiv P_{2u,t}u_{tt}+P_{3u,t}u_t^2+P_{1u,t}u_t+Q(t,u)=0,tag1 where uin D(N)subseteq Usubseteq V, yin[t_0,t_1]subset{bf R} and u_tequiv D_tuequivfrac{d}{dt}u, u_{tt}equivfrac{d^2}{dt^2}. par Assumptions: forall tin[t_0,t_1] and forall uin U_1, P_{iu,t}:U_ito V_i (i=1,dots,3) are linear operators and Q: [t_0,t_1]times U_1to V_1 is an arbitrary operator. The domain of N is defined as aligned D(N)=&{uin U: u|_{t=t_0}=varphi_1, u|_{t=t_1}=varphi_2, & u_t|_{t=t_0}=varphi_3, u_t|_{t=t_1}=varphi_4, varphi_iin U_1 (i=1,dots,4)},endaligned tag2 where U=C^2([t_0,t_1];U_1), V=C([t_0,t_1];V_1), and U_1,V_1 are linear normed spaces, U_1subseteq V_1. Moreover, for every tin(t_0,t_1) and g(t),u(t)in U_1 the functions P_{1u,t}g(t), P_{3u,t}g(t) are continuously differentiable and P_{2u,t}g(t) is twice continuously differentiable on (t_0,t_1). par Any function uin D(N) satisfying (1) is called the solution of (1). par The author constructs a certain functional which is called the first integral of equation (1) under conditions (2) if it doesn't depend on t for any solution u of the problem (1). Some concrete examples are given.