Integrals of an irrational two-frequency oscillator ($F=\varphi (R_1,R_2)$)

On the regular set of two uncoupled oscillators with irrational frequency ratio, every continuous globally defined integral of motion is constant on each fixed-action flat torus, because every orbit is dense there. A differentiable integral is consequently a function of the two actions, so its differential of a smooth map lies in their span. Local angle combinations can still give extra local integrals; they fail to be single-valued on the torus.