= Solution
Use <Gaussian units> and the <kinetic magnetohydrodynamics> equations
$$
\partial_t\mathbf B=\nabla\times(\mathbf u\times\mathbf B),\qquad
\rho D_t\mathbf u=-\nabla\cdot\mathbf P+\frac{(\nabla\times\mathbf B)\times\mathbf B}{4\pi},\qquad
\mathbf P=p_\perp\mathbf I+(p_\parallel-p_\perp)\mathbf b\mathbf b,
$$
together with species <continuity equations>, <quasineutrality> and the <drift-kinetic equation>. Here $p_\perp,p_\parallel$ sum the species <pressures>. Interpret the <thermal speed> denominator in the source as the species <mass> $m_\alpha$, so $v_{\mathrm{th}\alpha}^2=2T_{0\alpha}/m_\alpha$ and the <first adiabatic invariant> is $\mu=v_\perp^2/(2B)$.
The specified <Maxwell-Boltzmann distribution> has zero mean motion and <isotropic> <pressure>, $p_{\perp0}=p_{\parallel0}=p_0$. All equilibrium scalars depend only on $x$, while $\mathbf b_0\cdot\nabla x=0$ and $\mathbf b_0\cdot\nabla B_0=0$. Therefore the equilibrium streaming and mirror terms of the <drift-kinetic equation> vanish. The remaining terms vanish because $\mathbf u_0=0$ and $E_{\parallel0}=0$; the parallel electric constraint is also satisfied because the equilibrium parallel <pressure gradient> is zero. Species <continuity equations> and the induction equation are stationary, and <quasineutrality> holds by assumption.
It remains to check the perpendicular <force balance>. The radial <magnetic pressure> and <magnetic tension> give
$$
0=-\frac{dp_0}{dx}-\frac{d}{dx}\frac{B_0^2}{8\pi}-\frac{B_0^2}{4\pi R}
=\frac{p_0}{L_p}+\frac{B_0^2}{4\pi L_B}-\frac{B_0^2}{4\pi R}.
$$
This is precisely
$$
\boxed{\frac1R-\frac1{L_B}=\frac{4\pi p_0}{B_0^2L_p}=\frac\beta{2L_p}.}
$$
Thus the <Z-pinch> is a steady solution of the stated <kinetic magnetohydrodynamics> system. Its equilibrium <electric current> is supplied by the small species drifts implicit in this approximation; zero leading common flow does not require zero <electric current>.
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