= Solution
Let $q=(q_1,q_2,q_3)$ and $n=(1,1,1)$. Since
$$
q_1q_2+q_2q_3+q_3q_1
=\frac12\{(q\cdot n)^2-|q|^2\},
$$
the <Lagrangian> depends on $q$ only through $|q|^2$ and $(q\cdot n)^2$, and on the <velocity> through $|\dot q|^2$. Rotations about the $n$-axis preserve all three quantities, proving the invariance.
The <axially symmetric isotropic-mass Lagrangian> therefore conserves the axial <angular momentum>
$$
J_n=n\cdot(q\times\dot q)
=(q_2\dot q_3-q_3\dot q_2)
+(q_3\dot q_1-q_1\dot q_3)
+(q_1\dot q_2-q_2\dot q_1),
$$
up to an irrelevant normalization of $n$. Since $L$ has no explicit time dependence, <conservation of energy from time-translation invariance> gives the second conserved quantity
$$
\boxed{E=\frac12|\dot q|^2+\frac12|q|^2
+\alpha(q_1q_2+q_2q_3+q_3q_1).}
$$
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