Non-cooperative Nash Equilibrium in political game
You could add up the equations
\begin{align}
\delta(1-\underline{S})S_j=e_i(S_i+S_j)^2 \\
\delta(1-\underline{S})S_i=e_j(S_i+S_j)^2
\end{align}
to get
\begin{align}
\delta(1-\underline{S})(S_i+S_j)=(e_i+e_j)(S_i+S_j)^2
\end{align}
which simplifies to
\begin{align}
\delta(1-\underline{S})=(e_i+e_j)(S_i+S_j).
\end{align}
This gives you $(S_i+S_j)$. From the initial equations
\begin{align}
\delta(1-\underline{S})S_j=e_i(S_i+S_j)^2 \\
\delta(1-\underline{S})S_i=e_j(S_i+S_j)^2
\end{align}
you can also get
$$
\frac{S_j}{e_i} = \frac{(S_i+S_j)^2}{\delta(1-\underline{S})} = \frac{S_i}{e_j}, 
$$
which gives you
$$
\frac{S_j}{S_i} = \frac{e_i}{e_j}.
$$
Now you have both the sum and ratio of the variables $S_j,S_i$. Calculating their individual values should be a straightforward matter.