astrophysics - Exploiting symmetry in the axisymmetric Jeans equations

Background: If $f(mathbf{x},mathbf{v},t)$ is the distribution function of the stars in phase space and $n = int f,mathrm{d}^3mathbf{v}$ is the star density, then in the cylindrical coordinates $(R,varphi,z)$, the $z$-component of the second of the Jeans equations applied to an axisymmetric system is
$$renewcommand{expv}[1]{langle{#1}rangle}partial_t(nexpv{v_z}) + partial_R(nexpv{v_Rv_z}) + partial_z(nexpv{v_z^2}) + frac{nexpv{v_Rv_z}}{R} + npartial_zPhi = 0text{.}$$
In a steady state, the first term vanishes, so if the positive and negative terms of the $z$-component of velocity terms balance, then so do the second and fourth, in which case:
$$partial_z(nexpv{v_z^2}) = -npartial_zPhitext{,}$$
which we can then integrate.

If the the star and mass distributions are axisymmetric and symmetric about the equatorial ($z = 0$) plane, then the potential has even symmetry in $z$, $Phi(R,z) = Phi(R,-z)$, and so does $n$. Thus their derivatives with respect to $z$ are odd functions, and $Phi_{,z}(R,z) = -Phi_{,z}(R,-z)$ implies
$$frac{1}{n}int_{-z}^z nPhi_{,z'},mathrm{d}z' = 0text{,}$$
since the integrand has odd symmetry. Hence,
$$frac{1}{n}int_z^infty nPhi_{,z'},mathrm{d}z' = frac{1}{n}int_{|z|}^infty nPhi_{,z'},mathrm{d}z'text{.}$$