= Solution
Use the standard constant-density <incompressible flow> model, with conservative body force. Write $\mathbf u=(u,v,0)$ independent of $z$, and $\omega=v_x-u_y$. Differentiate the $y$-momentum <Euler equations> with respect to $x$ and subtract the $y$-derivative of the $x$-momentum equation. Mixed pressure derivatives and the curl of a conservative body force cancel, giving
$$
\omega_t+u\omega_x+v\omega_y+(u_x+v_y)\omega=0.
$$
The last term vanishes by <incompressibility>, so the material derivative of <vorticity> is zero. Thus \b[$D\omega/Dt=0$]. <Incompressibility> is important: inviscidness alone does not suffice; in a compressible barotropic planar flow the equation is $D\omega/Dt=-\omega\nabla\cdot\mathbf u$, conserving $\omega/\rho$ instead.
Take vortex strength $\kappa$ to denote <circulation>, positive counterclockwise. A <point vortex> at $\mathbf x_0$ generates
$$
\mathbf u(\mathbf x)=\frac{\kappa}{2\pi}\frac{J(\mathbf x-\mathbf x_0)}{|\mathbf x-\mathbf x_0|^2},\qquad J(x,y)=(-y,x).
$$
It has zero <vorticity> away from its core, with circulation $\kappa$ around the core. Each vortex moves with the other's regular induced <velocity>, not its own singular field. With $\mathbf d=\mathbf x_1-\mathbf x_2$,
$$
\dot{\mathbf x}_1=\frac{\kappa_2}{2\pi|\mathbf d|^2}J\mathbf d,\qquad \dot{\mathbf x}_2=-\frac{\kappa_1}{2\pi|\mathbf d|^2}J\mathbf d.
$$
Multiplication by the two strengths proves $\dot{\mathbf q}=\kappa_1\dot{\mathbf x}_1+\kappa_2\dot{\mathbf x}_2=0$. Also $\dot{\mathbf d}=(\kappa_1+\kappa_2)J\mathbf d/(2\pi|\mathbf d|^2)$, so $d|\mathbf d|^2/dt=2\mathbf d\cdot\dot{\mathbf d}=0$.
For equal strengths $\kappa$ and the specified initial positions, $\mathbf q=0$ and $|\mathbf d|=2a$. The <motion of two point vortices> is rotation with angular <velocity> $\Omega_v=\kappa/(4\pi a^2)$:
$$
\boxed{\mathbf x_1(t)=a(\cos\Omega_vt,\sin\Omega_vt),\qquad \mathbf x_2(t)=-\mathbf x_1(t).}
$$
Negative circulation reverses the rotation. If vortex strength is defined instead as the coefficient of $1/r$ <velocity>, absorb the factor $2\pi$ consistently into its definition.
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