= Solution
Set
$$
K(z)=-J\nabla W(z)=(W_y(z),-W_x(z)),\qquad L=\max\left\{1,\sup_z\|D^2W(z)\|_{\mathrm{op}}\right\}.
$$
Symmetry here means $W(-z)=W(z)$, so $\nabla W(0)=0$ and $K(0)=0$. The bounded <Hessian matrix> and the fact that $J$ is an <orthogonal matrix> give
$$
|K(z)-K(z')|\leq L|z-z'|,\qquad |K(z)|\leq L|z|.
$$
Thus the particle system is $\dot X^i=N^{-1}\sum_jK(X^i-X^j)$. On the configuration space with norm $\|X\|_{\max}=\max_i|X^i|$, its vector field has <Lipschitz constant> at most $2L$:
$$
|F_i(X)-F_i(Y)|\leq\frac L N\sum_j\left(|X^i-Y^i|+|X^j-Y^j|\right)\leq2L\|X-Y\|_{\max}.
$$
The <Picard-Lindelöf theorem> gives local existence and uniqueness. Linear growth and the <Gronwall inequality> give $\|X(t)\|_{\max}\leq e^{2Lt}\|X(0)\|_{\max}$, which rules out escape to infinity in finite time and extends the solution to every $t\geq0$.
The associated <empirical measure> is
$$
\boxed{\mu_t^N=\frac1N\sum_{i=1}^N\delta_{X^i(t)}.}
$$
Here each $\delta_{X^i(t)}$ is a <Dirac measure>. The finite first <absolute moment> follows from $\int|x|\,d\mu_t^N=N^{-1}\sum_i|X^i(t)|<\infty$. Pairing equal particle labels defines a <coupling>, so
$$
d_1(\mu_t^N,\mu_s^N)\leq\frac1N\sum_i|X^i(t)-X^i(s)|\longrightarrow0\quad(t\to s).
$$
Consequently
$$
\boxed{\mu^N\in C([0,\infty),\mathcal P_1(\mathbb R^2)).}
$$
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