Non-cooperative Nash Equilibrium in political game I have difficulties deriving the non-cooperative Nash Equilibrium of this problem.

The objective function is to maximize the expected total rent over the two periods, that is:

\begin{align}
\max_{S_i} (1-e_i S_i + P_i(S_i, S_j) \delta(1-\underline{S}))
\end{align}

with $P_i(S_i, S_j)=\frac {S_i}{S_i+S_j}$, and $i,j \in \{A,B\}$, and $e_A=1-d$ and $e_B=1+d$

$S_i$ denotes the quality of the service, $e_i$ is a parameter that captured the spending needs in $i$, $P_i(S_i, S_j)$ is the re-election probability, $\delta$ is the discount factor and $\underline{S}$ is the minimum service quality.

The solution of the Nash Equilibrium of the game is
\begin{align}
(S_A^*, S_B^*)=\Big(\frac{\delta (1+d)(1-\underline{S})}{4}, \frac{\delta (1-d)(1-\underline{S})}{4}\Big)
\end{align}

I have tried but I don't get that same solution. 

This is what I have done: 
I solve the maximization problem 
\begin{align}
\frac {\partial{U_i}}{\partial{S_i}} = -e_i+\dfrac{\delta(1-\underline{S})S_j}{(S_i+S_j)^2}\stackrel{!}{=}0
\end{align}

and get the best respond function for $i$ and $j$ respectively as follows
\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}

I impose symmetry $S_i=S_j=S^*$ to the reaction function and I get this: 
\begin{align}
\delta(1-\underline{S})S^*&amp;=e_i(2S^*)^2 \\
S^*&amp;=\dfrac{\delta(1-\underline{S})}{4e_i}
\end{align}

I am not sure though.
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.