We are given $f : \mathbb { R } ^ { 2 } \rightarrow \mathbb { C }$ and $g : \mathbb { R } ^ { 2 } \rightarrow \mathbb { R } ^ { 2 }$ two functions continuous and 1-periodic in each of their arguments, and $\omega \in \mathbb { R } ^ { 2 }$. We assume $x \neq 0$ (but close to 0). We consider a new unknown function $\tilde { \alpha } : \mathbb { R } \rightarrow \mathbb { R } ^ { 2 }$, of the form $$\tilde { \alpha } ( t ) = \alpha ( t ) + x h ( \alpha ( t ) ) ,$$ where $h : \mathbb { R } ^ { 2 } \rightarrow \mathbb { R } ^ { 2 }$ is an auxiliary function, 1-periodic in each of its arguments, of class $\mathscr { C } ^ { 1 }$ and of zero average, and which moreover, for some $\nu \in \mathbb { R } ^ { 2 }$, satisfies the equation $$\forall \theta \in \mathbb { R } ^ { 2 } , \quad d h ( \theta ) \cdot \omega + g ( \theta ) = \nu . \tag{4}$$
Determine $\nu$ as a function of $g$. In the case where the two components $g _ { 1 }$ and $g _ { 2 }$ of $g$ are trigonometric polynomials, deduce the existence of a solution $h$ of equation (4), which you will make explicit.