= Solution
Take the separate oscillator energies
$$
E_i=\frac12(y_i^2+\alpha_i^2x_i^2),\qquad i=1,2.
$$
They depend on disjoint canonical coordinate pairs, so
$$
\{E_1,E_2\}=0,\qquad\{E_i,H\}=0.
$$
Thus they are <first integrals in involution>. Their <differentials>
$$
dE_i=\alpha_i^2x_i\,dx_i+y_i\,dy_i
$$
are nonzero precisely when $(x_i,y_i)\ne(0,0)$ and have disjoint supports. Hence they are <functionally independent> on the specified open set.
To understand all further global <integrals of motion>, use the actions
$$
R_i=E_i/\alpha_i>0,\qquad
x_i=\sqrt{2R_i/\alpha_i}\sin\theta_i,\quad
y_i=\sqrt{2\alpha_iR_i}\cos\theta_i.
$$
Direct differentiation gives $dx_i\wedge dy_i=d\theta_i\wedge dR_i$, so these are <action-angle variables> with
$$
H=\alpha_1R_1+\alpha_2R_2,\qquad
\dot R_i=0,\quad\dot\theta_i=\alpha_i.
$$
When $\alpha_2/\alpha_1$ is irrational, every orbit is dense in its fixed-action two-dimensional <flat torus>. One elementary proof samples the orbit at times $2\pi j/\alpha_1$: the first angle returns, and the second undergoes an <irrational rotation>. These samples are dense in the second circle. Allowing a fixed additional time supplies any desired first angle, proving density in the whole <flat torus>.
A continuous, globally defined <integral of motion> must have the same value on an orbit and its closure. Therefore it is constant on every fixed-action <flat torus>. A $C^1$ such integral has the form $F=\phi(R_1,R_2)$, with $\phi$ $C^1$, by evaluating $F$ at a fixed choice of angles. Hence
$$
\boxed{dF=\phi_{R_1}\,dR_1+\phi_{R_2}\,dR_2.}
$$
It cannot be independent of $E_1,E_2$. This proves \b[there are at most two globally independent differentiable integrals]. If a third global integral were independent at a point on an excluded coordinate plane, independence would persist in a neighbourhood and hence at nearby points of the regular open set, again a contradiction.
The word global matters: on an angular chart, $\alpha_2\theta_1-\alpha_1\theta_2$ is a local <integral of motion> independent of the actions. It is not single-valued on the <flat torus> when the frequency ratio is irrational. Thus the assertion would be false for arbitrary local, rather than global, integrals.
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