I have a space with dimension $N$ and $n<N$ vector fields $\mathbf{F}_j$. They commute: $$ \left[ \mathbf{F}_j , \mathbf{F}_k \right] = 0 $$ Then I have $n$ scalar fields $\varphi_j$, which satisfy this condition: $$ \mathcal{L}_{F_j} \varphi_k = \mathcal{L}_{F_k} \varphi_j $$ where $\mathcal{L}_F$ is the Lie derivative along $\mathbf{F}$. This condition is somehow similar to the one for closed one-forms, however, $n$ is less than the dimension of the space, $N$, so I do not expect to have a differential form, nor that the $\varphi_j$ are the derivatives of the same scalar field.
What is the name of this condition? Is there a reference where this condition is discussed and the consequences explained?