= Solution
The potential factorizes as
$$
U(\phi)=\phi^2(\phi^2-4)^2.
$$
It is even and nonnegative, with three degenerate vacua at $\phi=-2,0,2$. Its two intervening maxima occur at $\phi=\pm2/\sqrt3$ and have height $256/27$.
A finite-energy static solution in one dimension satisfies
$$
\phi''=U'(\phi).
$$
Multiplication by $\phi'$ and use of the vacuum boundary conditions gives the <first integral>
$$
\frac12(\phi')^2=U(\phi).
$$
Any path from a negative vacuum to a positive vacuum must pass through the intermediate vacuum $\phi=0$. There both $U$ and $\phi'$ vanish. The <Picard-Lindelof theorem> then forces a solution reaching $(\phi,\phi')=(0,0)$ at finite $x$ to remain there. Equivalently, the first-order orbit approaches $\phi=0$ only as $|x|\to\infty$. This <intermediate-vacuum obstruction to a kink> means that a single kink cannot connect $-2$ to $2$; it splits into two elementary kinks at infinite separation.
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