Background and Statement of the Problem
We consider the problem in $\mathbb{R^2}.$ As the title states, we are interested in the following equation involving the Hamiltonian of the point vortex system, $H^{2,\Omega},$ and the positions of a pair of point vortices in Lagrangian coordinate, $x_1(t)$ and $x_2(t).$
$$\Omega_j \frac{\text{d}x_j}{\text{d}t} = J\nabla_jH^{2,\Omega} (x_j). \tag{1} \label{1}$$
Here, $x_j = \begin{bmatrix} x_{j,1} \\ x_{j,2}\end{bmatrix},$ for $j=1,2.$ $J = \begin{bmatrix} 0 & 1 \\ -1 & 0 \end{bmatrix}$ , and $\nabla_j = \begin{bmatrix} \partial_{x_{j,1}} \\ \partial_{x_{j,2}} \end{bmatrix}.$
The Hamiltonian for the system is given by the following formula:
$$H^{2,\Omega}(x_j) = \frac{1}{2} \sum^{2}_{\begin{matrix}k,l=1 \\ k \neq l\end{matrix}} \Omega_k \Omega_l G(x_{j,k} , x_{j,l}), $$
where $G(x_{j,1} , x_{j,2}) = \frac{1}{2\pi} \ln| x_{j,1} - x_{j,2} |^{-1}$ is just the Green's function for the Laplace Equation in the whole space.
We set the constants $\Omega_i$ for this problem as follows, $\Omega_1 = \pm \Omega_2 = 1. $ Thus the Hamiltonian simplifies to the Green's function.
$$H^{2,\Omega}(x_j) = \frac{1}{2\pi} \ln| x_{j,1} - x_{j,2} |^{-1}. $$
We can then can rewrite \eqref{1} as the following pair of vector equations:
\begin{align} \frac{\text{d} x_1}{\text{d}t} = \pm J \nabla_1 \Big{(} \frac{1}{2\pi} \ln|x_1 - x_2|^{-1} \Big{)}, \\ \frac{\text{d} x_2}{\text{d}t} = J \nabla_2 \Big{(} \frac{1}{2\pi} \ln|x_1 - x_2|^{-1} \Big{)}. \end{align}
My Work
The spatial derivative on the right hand side serves only to rewrite the log function into a fraction involving absolute values. We're left with a system of 4 ODEs with respect to the time variable, which are written as follows:
$$ \begin{align} \frac{\text{d} x_{1,1}}{\text{d}t} & = \pm \Big{(} -\frac{1}{2\pi} \frac{(x_{1,2} - x_{2,2})}{|x_1 - x_2|^2} \Big{)} \\ \frac{\text{d} x_{1,2}}{\text{d}t} & = \pm \Big{(} \frac{1}{2\pi} \frac{(x_{1,1} - x_{2,1})}{|x_1 - x_2|^2} \Big{)} \\ \frac{\text{d} x_{2,1}}{\text{d}t} & = \Big{(} \frac{1}{2\pi} \frac{(x_{1,2} - x_{2,2})}{|x_1 - x_2|^2} \Big{)} \\ \frac{\text{d} x_{2,2}}{\text{d}t} & = \Big{(} -\frac{1}{2\pi} \frac{(x_{1,1} - x_{2,1})}{|x_1 - x_2|^2} \Big{)}. \end{align}$$
I have never come across a system of ODEs such as this before, and am at a complete loss as to how to solve them. Any hints or references to texts with this or a similar problem are very much appreciated. Thank you.
Edit 1.
The above equations are perhaps better represnted in the following vector form:
$$ \begin{align} \frac{\text{d} x_{1}}{\text{d}t} & = \pm \Big{(} - J \frac{x_1 - x_2}{|x_1 - x_2|^2} \Big{)}. \\ \frac{\text{d} x_{2}}{\text{d}t} & = \pm \Big{(} J \frac{x_1 - x_2}{|x_1 - x_2|^2} \Big{)}. \end{align} $$
It is then worth noting the main differences between the plus and minus cases for $x_1$. In the plus case, we get
$$ \frac{\text{d} x_{1}}{\text{d}t} + \frac{\text{d} x_{2}}{\text{d}t} = 0, $$
which I believe corresponds to the vortices revolving together around some centre. In the minus case, we get
$$ \frac{\text{d} x_{1}}{\text{d}t} - \frac{\text{d} x_{2}}{\text{d}t} = 0, $$
where the two point vortices travel parallel to each other.