\alpha_N(r) > 0$ for all $r\in[0,\infty)$.

4.  $\psi\in C_c^\infty (\mathbb R^2)$ is compactly supported, $\psi \geq 0$, $\partial_L \psi(\lambda, 0) = 0$ for $\lambda \in \mathbb R$, and $\psi(0, L) = 1$ on an open neighbourhood of the set $\{L = L_N(x,v) \, | \, (x,v) \in \mathrm{supp}(f_N)\}$, where $L_N := \varrho v^3$ is the $z$-component of the Newtonian angular momentum.

There exist ansatz functions $\phi \in C_c^2(\mathbb R)$ and $\psi\in C_c^\infty((-1/2, 1/2) \times \mathbb R)$ satisfying the upper conditions.

Consider the polytropes $\phi(E) = [E_0-E]_+^k$ for $k\in[2,7/2)$. Condition (1) is clearly satisfied. Condition (2) is also satisfied, cf. [@bfh86; @rr00].

By the same proof as for [@akr11 Lemma 7.1] it can be shown that the third condition is satisfied for polytropes with exponent $k$ sufficiently close to $7/2$. To this end one uses the equation $\Delta U_N = 4\pi (1-q^2) \rho_N$ instead of $\Delta U_N = 4\pi \rho_N$. Then merely the constant $4\pi$ has to be replaced by $4\pi(1-q^2)$ in the proof of [@akr11 Lemma 7.1]. It is essential that $1-q^2 > 0$, the precise value is however irrelevant for the argument.

The reduced system of equations {#sect_red_sys}
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Before the reduced system of equations is presented some notation 