On the geometry of point-transformation invariant class of third-order ordinary differential equations (Q2388178)

In this paper, third-order differential equations of the reduced form \[ y'''=Py''{y'}^2+Qy'' y'+Ry''+S{y'}^5+L{y'}^4+K{y'}^3+M{y'}^2+Ny'+T\tag{\(*\)} \] are analyzed, where the coefficients are functions in \((x,y)\). These equations constitute the expression over an appropriate coordinate system of the class of third-order equations shown by \textit{V. V. Dmitrieva} [Mat. Zametki 70, No. 2, 195--200 (2001; Zbl 1031.34037)] to be invariant with respect to non-degenerate transformations \(\widetilde{x} =\widetilde{x}(x,y),\) \(\widetilde{y} =\widetilde{y}(x,y).\) In particular, they contain the third-order Painlevé equations. The previous reduced equation is invariant with respect to transformations of the form \(\widetilde{x}=\widetilde{x}(x),\) \(\widetilde{y} =\widetilde{y}(x) +y.\) Using pseudo-vector fields, the behaviour of the coefficients of (\(*\)) is studied under the preceding transformations. Imposing that three of these pseudo-vector fields vanish, (\(*\)) can be further reduced to \(y'''=Ry''+Ny'+T.\) The analysis of the defining equations of the latter reduced expression leads to two further pseudo-vector fields, and it is shown that these vanish if and only if the Lie algebra of the point symmetries is the seven dimensional algebra spanned by the fields \[ \left\langle \frac{\partial}{\partial x},x\frac{\partial}{\partial x} +\frac{\partial}{\partial y},\frac{\partial}{\partial y},x\frac{\partial }{\partial y},y\frac{\partial}{\partial y},x^{2}\frac{\partial}{\partial y},\frac{x^{2}}{2}\frac{\partial}{\partial x}+xy\frac{\partial}{\partial y}\right\rangle. \] Therefore, the five vanishing pseudo-vector fields characterize the equations having the normal form \(y^{\prime\prime\prime}=0\), according with the result obtained by \textit{A. Schmucker} and \textit{G. Czichowski} [J. Lie Theory 8, No. 1, 129--137 (1998; Zbl 0897.34013)].