An extension of the representation space of the group admitted by a system of ordinary differential equations (Q2387987)

The concept of a symmetry of the system of first-order equations \(\dot {x}_i = f_i(t,x),\;i = 1,n \), is extended by the inclusion of parameters of the system as variables to be considered in the application of the infinitesimal transformation generated by the symmetry. Thus a symmetry \(\Gamma = \xi_i\partial_{x_i} + \eta_i\partial_{\alpha_i} \) for the system \(\dot {x}_i = f_i(t,x,\alpha),\;i = 1,n \), is considered. (There is no need to introduce a \(\tau\partial_t \).) Even if the system contains not parameters, these can be introduced by replacing numerical values by parameters. Indeed this is a standard procedure to extend an equation (or system) which may arise under particular circumstances to a class of equations. The particular thrust of this paper is in the direction of holonomic constraints -- the parameters could be functions as in the case of control theory -- and indeed partial holonomic constraints, in other words, first integrals and configurational invariants. A couple of examples are given. The second provides a factoring of the generalised Ermakov equation, \(y ^3\ddot {y} - fy ^ 4 -\dot {g}y ^ 2+g ^2= 0 \), without the need to resort to nonlocal symmetries as in an earlier treatment of this equation. This prompts two questions. Does the approach by Kusyumov obviate the need to consider nonlocal symmetries in such systems? If there be not the case, are even richer results available with the inclusion of nonlocal symmetries? This is a short paper and in translation very clearly written. Together with the two questions this invites attention of a student.