I have a specific smooth function $F(x,y,z) = 0$ which implicitly defines a function $\widehat{z}(x,y)$.
In my analysis of this function's properties, the term $$M(x,y) \equiv -\frac{\widehat{z}_y}{\widehat{z}_x}$$ (that is, the ratio of partial derivatives of $\widehat z$) appears frequently. This quantity is evidently the slope of the level curve of $F$ obtained by intersecting with the plane $z = \widehat{z}(x,y)$ (Geometric meaning of partial derivatives $z_y/z_x$).
In particular, I've found a curious property: the mixed partials of $\log{M}$ for my function are identically zero, which means that $M(x,y) = \alpha(x)\beta(y)$ for some functions $\alpha$ and $\beta$.
My question is:
Question: What does this factorization mean geometrically for the function $F$ and its level curves?
I would think we can define the level curve in terms of an integral over $M(x,y)$, the slope, but when I write it out, I don't see any simplification to be made when the integrand factors as a product. Is there something we can learn about the level curves in any case?
Edit: I note also that
$$M(x,y) \equiv -\frac{z_y}{z_x} = \alpha(x) \beta(y) \quad \Rightarrow\quad \frac{z_y(x,y)}{\alpha(x)} = \frac{-\beta(y)}{z_x(x,y)}$$ but I don't know if this separation leads anywhere.