Let $G(x,y)$ be a power series in two variables such that it can be rewriting in a such way $$ G(x,y)=a_0+a_1 x P_1(y)+a_2 x^2 P_2(y)+\cdots+a_n x^n P_n(y)+\cdots $$ where $P_k(y)$ is a polynomial in $y$ of degeree $k$ and all $a_k \neq 0.$ Let us call such series as $x$-balansed power series.
Suppose $G(x,y)$ is given as a closed expression not in expanded way, for example $$ G_1(x,y)=\frac{e^{xy}}{1-y^2} \text{ or } G_1(x,y)=\frac{e^{x}}{1-y} $$ Then $G_1(x,y)$ has a balansed series since $$ G_1(x,y)=\frac{e^{xy}}{1-y^2}=1+xy+ {x}^{2}\left( \frac 1 2 \,{y}^{2}+1 \right)+ {x}^{3} \left( \frac 1 6\,{y}^{3}+y \right) + \cdots+ $$ and $G_2(x,y)$ has no balansed series since
$$ G_2(x,y)=\frac{e^{x}}{1-y}=1+y+ x\left( {y}^{4}+{y}^{3}+{y}^{2}+y+1 \right) +{y}^{2}+\cdots $$
Question What conditions the function $G(x,y)$ must satisfy in order its series to be balanced?
One set of functions with such properties is obvious: $G(x,y)=F(xy)$ i.e. $G(x,y)$ is a function of one variable $xy$.
Is there any good analytical description of such functions?