On the structure of the equivalent differential systems and their reflecting integrals (Q1684179)

Consider in \(\mathbb{R}^n\) the differential equation \[ {dx\over dt}= X(t,x),\tag{\(*\)} \] where \(X\) is sufficiently smooth. Let \(\phi(t,t_0,x_0)\) be the solution of \((*)\) satisfying \(\phi(t_0,t_0,x_0)= x_0\). The function \(F(t,x)= \phi(-t,t,x)\) is called the reflecting function of \((*)\). Two systems \((*)\) are called equivalent if they have the same reflecting function. The author is interested in characterizing systems \[ {dx\over dt}= Y(t,x) \] which are equivalent to \((*)\). His main result characterizes \(Y(t,x)\) in the form \[ Y(t,x)= X(t,x)+ \sum^n_{i=1} \alpha_i(t,u)\Delta_i(t,x), \] where \(\alpha_i\) is a scalar function odd and smooth in \(t\), \(U=c\) is the general integral of \((*)\) and \(\Delta_i\) is a function satisfying \(\Delta_t+ \Delta_x X-X_x\Delta= 0\).