On special solutions to the Ermakov-Painlevé XXV equation (Q6549270)

Let \(F\) be an ordinary differential field of characteristics zero with differentiation \('=d/dz\). Denote by \(EP_{F}\) a family of differential equations of the form\N\[\N(12w+b(z))w''=15w^{,2}-h_{0}(z)w'-8w^{3}-h_{1}(z)w^{2}-h_{2}(z)w-h_{3}(z),\N\]\Nwhere \(b,h_{0},\ldots,h_{3}\in F.\) Similarly, introduce the notation \(\mathcal{L_{\textrm{F}}}\) and \(E_{F}\) for the families of equations \N\[\Ny'''+p(z)y''+q(z)y'+r(z)y=0 \text{ and } y''+\Phi(z)y=ay^{-3},\N\] \Nrespectively. The paper discusses the possibility of reduction of equations from \(EP_{F}\) to equations from \(\mathcal{L_{\textrm{F}}}\) and \(E_{F}\). In other words, there are selected such non-trivial subsets of equations in the family \(EP_{F}\), which can be differently-rationally maped to the corresponding subsets in \(\mathcal{L_{\textrm{F}}}\) or \(E_{F}\). For example, as it is shown in the paper, a one-parameter subset from \(EP_{F}\)\N\[\N(3(k^{''}+2w^{''})+2(2k+w)(k+2w))(k+2w)-\dfrac{15}{4}(k^{'}+2w^{'})^{2}=0\:(k\in F),\N\]\Nis maped on the \(E_{F}\) thus\N\[\Nu^{''}=\dfrac{1}{4}ku+\dfrac{\lambda}{72}u^{-3}(k\in F,\lambda=\mathrm{const}).\N\]\NHowever, the selection of the corresponding subset for the case \(\mathcal{L_{\textrm{F}}}\) is a non-trivial task, despite the presence of a differential-regular map \(\mathcal{L_{\textrm{F}}}\) in \(EP_{F}\). The authors describe their ideas to solve this problem.