  We study the singularly perturbed nonlinear energy critical Choquard equation
\begin{equation*} -{\Laplace u}\qty({x}) -{\alpha}
\int_{\R^N}\frac{u^p\qty(y)}{\abs{x-y}^{\lambda}}\odif{y} u^{p-1}\qty({x})
-\eps k\qty(x)u^{\frac{N+2}{N-2}}\qty(x)=0, \qquad x\in\R^N, \end{equation*}
where $N\geq 3$, $0<\lambda<N$, $\lambda\leq 4$, $p=\frac{2N-\lambda}{N-2}$,
$\alpha = \frac{
  N\qty({N-2})\fct{\Gamma}{N-\frac{\lambda}{2}} }{
  \pi^{\frac{N}{2}}\fct{\Gamma}{\frac{N-\lambda}{2}} }$,~ and $k$ is a positive
function. By making use of a Lyapunov-Schmidt reduction argument, for
sufficiently small $\eps>0$, we construct solutions of the form \begin{align*}
u_{\eps}\qty(x)=U_{\mu_{\eps},\xi_{\eps}}\qty(x)\qty(1+\O\qty(\eps)),
\end{align*} where $U_{\mu_{\eps},\xi_{\eps}}$ is a positive solution of the
unperturbed equation \begin{equation*}
  -{\Laplace u}\qty({x})
  -{\alpha}
  \int_{\R^N}\frac{u^p\qty(y)}{\abs{x-y}^{\lambda}}\odif{y}=0,\qquad x\in\R^N.
\end{equation*}
