We analyze the classical equations of motion for a particle moving in thepresence of a static magnetic field applied in the $ z $ direction, whichvaries as $ {1\over{x^2}} $. We find the symmetries through Lie's method ofgroup analysis. In the corresponding quantum mechanical case, the method ofspectrum generating $su(1,1)$ algebra is used to find the energy levels for theSchroedinger equation without explicitly solving the equation. The Lie pointsymmetries are enumerated. We also find that for specific eigenvalues thevector 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 generatorsdo not close.
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机译：我们分析了粒子在$ z $方向上施加的静态磁场的作用下运动的经典运动方程，其变化为$ {1 \ over {x ^ 2}} $。我们通过李的群分析方法找到对称性。在相应的量子力学情况下，使用频谱生成$ su（1,1）$代数的方法来找到Schroedinger方程的能级，而无需明确求解方程。列举了李点对称性。我们还发现，对于特定的特征值，vector字段包含$ {1 \ over {x}} {{\ p} \ over {\ px}} $和$ {1 \ over {x ^ 2}} {{\ p} \ {\ p {x}}} $类型的条款和生成器的有限Lie乘积不会关闭。
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