  In this paper, we study the following Hamiltonian Choquard-type elliptic
systems involving singular weights \begin{eqnarray*}
  \begin{aligned}\displaystyle
  \left\{ \arraycolsep=1.5pt
  \begin{array}{ll}
  -\Delta u + V(x)u = \Big(I_{\mu_{1}}\ast
\frac{G(v)}{|x|^{\alpha}}\Big)\frac{g(v)}{|x|^{\alpha}} \ \ \ & \mbox{in} \
\mathbb{R}^{2},\\[2mm]
  -\Delta v + V(x)v = \Big(I_{\mu_{2}}\ast
\frac{F(u)}{|x|^{\beta}}\Big)\frac{f(u)}{|x|^{\beta}} \ \ \ & \mbox{in} \
\mathbb{R}^{2},
  \end{array}
  \right.
  \end{aligned} \end{eqnarray*} where $\mu_{1},\mu_{2}\in(0,2)$, $0<\alpha \leq
\frac{\mu_{1}}{2}$, $0<\beta \leq \frac{\mu_{2}}{2}$, $V(x)$ is a continuous
positive potential, $I_{\mu_{1}}$ and $I_{\mu_{2}}$ denote the Riesz potential,
$\ast$ indicates the convolution operator, $F(s),G(s)$ are the primitive of
$f(s),g(s)$ with $f(s),g(s)$ have exponential growth in $\mathbb{R}^{2}$. Using
the linking theorem and variational methods, we establish the existence of
solutions to the above problem.
