The third independent conserved quantity and its symmetry of the two-dimensional anisotropic harmonic oscillator
二维各向异性谐振子的第三个独立守恒量及其对称性

The energy and the two partial energies of two-dimensional anisotropic harmonic oscillator are conserved quantities,but only two of them are independent.The system possesses the third independent conserved quantity when the ω1/ω2 is a rational number.The extended Prelle-Singer method is used to find the third independent conserved quantity for the five typical two-dimensional anisotropic harmonic oscillators.The Noether symmetry and the Lie symmetry of the third independent conserved quantities are also discussed.