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It is well known that the symmetry of spacetime with respect to rotating about some axis results in angular momentum along that axis being conserved, by Noether's theorem. E.g. rotational symmetry with respect to the $x$ axis implies $J_x$ is conserved.

Is there some symmetry that corresponds with total angular momentum $|\vec{J}|$ (or $J^2$ if you like) being a conserved quantity?

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If the Lagrangian of a system is invariant under rotations with respect to an arbitrary direction $\mathbf{n}$ in space, the corresponding conserved quantity is the component $\mathbf{n} \cdot \mathbf{J}$ of the total angular momentum in this direction. If this invariance holds for any direction $\mathbf{n}$ in space, the total angular momentum vector $\mathbf{J}$ of the system is conserved and consequently also $\mathbf{J}^2$.

The relevant symmetry group is $\rm SO(3)$ (or $\rm SU(2)$ for systems with half-integer spin).

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  1. The symmetry Lie group for angular momentum is ${\rm Spin}(3)$.

  2. If $\delta_{J_x}, \delta_{J_y}, \delta_{J_z}$ are the infinitesimal symmetries of the action $S$, generated by the angular momentum $J_x$, $J_y$, $J_z$, for rotations around the $x$, $y$, and $z$ axis, respectively; then $$2|\vec{J}|\delta_{|\vec{J}|}~=~ \delta_{J^2}~=~2J_x\delta_{J_x}+2J_y \delta_{J_y}+2J_z \delta_{J_z}\tag{1}$$ is the infinitesimal symmetry generated by the angular momentum square $J^2$. Eq. (1) follows e.g. from my Phys.SE answer here.

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