Interpretation of this PDE solving method needed

I am working on an open source project that deals with the usage of lie groups to solve first order differential equations. For my reference, I am using this paper by Maple (It is there on this website http://arxiv.org/abs/gr-qc/9607037v1).

Basically the idea is to solve this PDE, $$\frac{\partial \eta}{\partial x} + (\frac{\partial \eta}{\partial y} - \frac{\partial \xi}{\partial x})*h - \frac{\partial \eta}{\partial x}*h^{2} - \xi*\frac{\partial h}{\partial x} - \eta*\frac{\partial h}{\partial y} = 0$$ for $\xi$ and $\eta$ where $h$ is a known function in $x$ and $y$

One of the assumptions made is $$\eta = f(y)$$ and $$\xi = g(x)$$

Applying it, the PDE simplifies to,

$$(\frac{df}{dy} - \frac{dg}{dx})*h - f*\frac{\partial h}{\partial x} - g*\frac{\partial h}{\partial y} = 0$$

Now the paper says, subdivide the equation into subexpressions involving only one of {f, g}. I first though that it meant, solving $$\frac{df}{dy}*h - f*\frac{\partial h}{\partial x} = 0$$ and $$-\frac{dg}{dx}*h - g*\frac{\partial h}{\partial y} = 0$$ individually. However that doesn't seem to be this case as, the example given in the paper doesn't satisfy this condition. I would like some help in interpreting what "subdivide the equation into subexpressions involving only one of {f, g}".

The mentioned method is on page 9 of the paper which is in the above mentioned website, if anyone is interested.

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