Weakly irregular quasiperiodic solutions of nonlinear Pfaff systems

We consider the nonlinear quasiperiodic Pfaff system

$$\frac{{\partial x}}{{\partial t_j }} = F^{(j)} (t,x) + G^{(j)} (t,x)(j = 1,...,m).$$

Let K ^(j) be a frequency basis with respect to t _j of the functions F ⁽¹⁾,...,F ^(m), and let L ^(j) be a frequency basis with respect to t _j of the functions G ⁽¹⁾,...,G ^(m). Suppose that the set K ^(j) ∪ L ^(j) of numbers is rationally linearly independent. We obtain necessary and sufficient conditions for the existence of quasiperiodic solutions with frequency bases L ⁽¹⁾,..., L ^(m).