Symmetry analysis for a charged particle in a certain varying magnetic field

We analyze the classical equations of motion for a particle moving in the presence of a static magnetic field applied in the $z$ direction, which varies as ${1\over{x^2}}$. We find the symmetries through Lie's method of group analysis. In the corresponding quantum mechanical case, the method of spectrum generating $su(1,1)$ algebra is used to find the energy levels for the Schroedinger equation without explicitly solving the equation. The Lie point symmetries are enumerated. We also find that for specific eigenvalues the vector field contains ${1\over{x}} {{\p}\over{\p x}}$ and ${1\over {x^2}} {{\p}\over{\p {x}}}$ type of terms and a finite Lie product of the generators do not close.