= Solution
Set $f(A)=\mu_0A+\alpha A^3-A^5$ and $V(A)=\mu_0A^2/2+\alpha A^4/4-A^6/6$. Multiplication of the stationary <reaction-diffusion equation> by $A_x$ gives the first integral $A_x^2/2+V(A)=E$. A regular <front> with finite endpoint limits has $A_x\to0$ there, and its nonzero endpoint must satisfy $f(C)=0$. Since $V(0)=0$, the first integral gives $E=0$ and $V(C)=0$. Write $q=C^2>0$. The two endpoint conditions are
$$
\mu_0+\alpha q-q^2=0,\qquad
\frac{\mu_0}{2}+\frac{\alpha q}{4}-\frac{q^2}{6}=0.
$$
Eliminating $\mu_0$ gives the <Maxwell balance for a scalar reaction-diffusion front>:
$$
\boxed{q=C^2=\frac{3\alpha}{4},\qquad \mu_0=-\frac{3\alpha^2}{16}.}
$$
This is a necessary condition for a stationary front, because the two spatially homogeneous states must have equal potential. It is also sufficient here. With these values, $-2V(A)=A^2(A^2-q)^2/3$. For the increasing positive front choose
$$
A_0'=\frac{1}{\sqrt3}A_0(q-A_0^2),\qquad
s=A_0^2,\qquad s'=\frac{2}{\sqrt3}s(q-s).
$$
Integrating this logistic equation and writing $\beta=2q/\sqrt3=\sqrt3\alpha/2$ gives
$$
\boxed{A_0^2(\xi)=\frac{q}{1+q e^{-\beta\xi}},\qquad \xi=x-X_0.}
$$
The positive constant multiplying the exponential is arbitrary and can be absorbed into $X_0$; choosing it to be $q$ gives the printed normalization exactly. The profile tends to zero and $q$ at the two ends and its first-order equation implies $A_0''+f(A_0)=0$. Its negative is the front to $-\sqrt q$, with the same squared profile.
Differentiation of the stationary equation produces the translation mode
$$
\mathcal L A_0'=0,\qquad
\mathcal L=\partial_x^2+\mu_0+3\alpha A_0^2-5A_0^4.
$$
The operator $\mathcal L$ is <self-adjoint>. Twice applying <integration by parts> gives, for $R$ with vanishing boundary Wronskian,
$$
\int_{-\infty}^{\infty}A_0'\mathcal LR\,dx
=\left[A_0'R'-A_0''R\right]_{-\infty}^{\infty}
+\int_{-\infty}^{\infty}R\mathcal LA_0'\,dx=0.
$$
Bounded $R,R'$ suffice, because the derivatives of the front decay exponentially. This proves the identity as a <Fredholm solvability condition> associated with translation invariance.
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