We are given a one form $\omega=x dy + dz \in \mathbb{R}^3$ and asked to give conditions on a vector field $X$ such that $\mathcal{L}_X \omega = 0$ and then give examples of such vector fields. My approach:
$$\mathcal{L}_X \omega = \mathcal{L}_X (x) dy + x * d(\mathcal{L}_X (y)) + d(\mathcal{L}_X (z))$$ where $\mathcal{L}_X (dy) = d * (\mathcal{L}_X (y))$ for example, because of commutation of $d$ with $\mathcal{L}_X$. But if we take $X = f_1 \frac{\partial}{\partial x} + f_2 \frac{\partial}{\partial y} + f_3 \frac{\partial}{\partial z}$ as our arbitrary vector field, then we have $\mathcal{L}_X (x) = f_1$, $\mathcal{L}_X (y) = f_2$ and $\mathcal{L}_X (z) = f_3$ so that $$\mathcal{L}_X \omega = f_1 dy + x df_2 + df_3$$. If this is zero then what am I saying here? It cannot be $f_1 = 0$, $d f_2 = 0$ and $d f_3 = 0$ right? I welcome any and all pointers on this.