Mass-weighted oscillator symmetries

For independent coordinates with $L={\sum }_i{m}_i({\dot{x}}_i^2-{\omega }^2{x}_i^2)/2$, the variables $X_i=\sqrt{m_i}{x}_i$ give an isotropic oscillator even at unequal masses. Rotations of the $X_i$ are continuous symmetries. Their conserved Noether charges $X_i{P}_j-X_j{P}_i$ therefore predate any equality of physical masses. Equal masses alone cannot create new invariances of arbitrary additional interactions.