On equivalence of singularities of second order linear differential equations by point transformations (Q2053315)

The author considers a second-order linear ordinary differential equation \[ \dfrac{d^2y}{dx^2}+a_1(x)\dfrac{dy}{dx}+a_0(x)y=0, \quad (x,y)\in\Omega\times\mathbb C, \tag{1}\] on a complex domain \(\Omega\subset\mathbb C\), the coefficients \(a_0(x)\) and \(a_1(x)\) are meromorphic on \(\Omega\), together with its companion system \[\dfrac{dv}{dx}=\begin{pmatrix}0&1\\ -a_0(x)&-a_1(x)\end{pmatrix}v, \quad v=\left(y,\dfrac{dy}{dx}\right) ^t, \quad (x,v)\in\Omega\times\mathbb C^2.\tag{2}\] Given a linear point transformation of both the independent variable \(x\) and of the unknown function \(y\) \[\widetilde{x}=\phi(x),\quad\widetilde{y}=t(x)y \] with \(\phi:\Omega\rightarrow\Omega\) an analytic diffeomorphism and \(t(x)\) either a non-vanishing analytic, or a non-zero meromorphic, function, the corresponding equations for \(y\) and \(\widetilde{y}\) are called analytically (resp. meromorphically) equivalent. With respect to the transformation (2), the question of the local classification of the singularities of Eq. (1) is asked. The author proves in particular that, under a convenient non-degeneracy condition, the equations for \(y\) and \(\widetilde{y}\) are analytically equivalent if and only if the associated companion systems are analytically equivalent as systems. He also provides the Lie algebras of the analytic linear infinitesimal symmetries of the singularities.