= Solution
Set $A_D=S_D\Lambda^\epsilon/(2\pi)^D>0$. The one-loop flow is $\mathcal B(g)=-\epsilon g+(n-2)A_Dg^2$. For $\epsilon>0$ and $n>2$, its physical fixed points are
$$
\boxed{g_0=0,\qquad g_*=\frac{\epsilon}{(n-2)A_D}=\frac{(2\pi)^D\epsilon}{(n-2)S_D\Lambda^\epsilon}.}
$$
The zero point attracts small positive couplings, whereas the nonzero point is the separatrix between the low-temperature weak-coupling flow and the strong-coupling regime. This is a perturbative description for small $\epsilon$; extrapolation of its nonzero root to large $\epsilon$ is not controlled.
At $\epsilon=0$ and $n>2$, the only finite one-loop fixed point is $g=0$, a double root. Positive $g$ flows to strong coupling. For $n=2$ the loop coefficient vanishes: at $\epsilon>0$ only $g=0$ is fixed, while at $\epsilon=0$ every $g$ is fixed in this perturbative spin-wave description. The compact two-component model also admits vortices; their <Berezinskii–Kosterlitz–Thouless transition> is absent from a calculation retaining only the smooth tangent modes. Thus a zero beta function here does not exclude that transition. The formula should not be interpreted as a continuum result for the discrete $n=1$ Ising target.
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