= Solution
The axis-regular <separation of variables> expansion for an axisymmetric harmonic potential is
$$
\Phi(r,x)=\sum_{n=0}^\infty(A_nr^n+B_nr^{-n-1})P_n(x).
$$
The radial powers solve $r^2R''+2rR'-n(n+1)R=0$, while the regular angular solutions are <Legendre polynomials>. For an exterior solution tending to zero at infinity, retain only the negative radial powers. The boundary polynomial decomposes as $1+x^2=\tfrac43P_0(x)+\tfrac23P_2(x)$. Therefore
$$
\boxed{\Phi(r,x)=\frac{4a}{3r}+\frac{2a^3}{3r^3}P_2(x)
=\frac{4a}{3r}+\frac{a^3}{3r^3}(3x^2-1),\quad r\ge a.}
$$
It has the required boundary value and decay. Uniqueness follows from the <maximum principle for harmonic functions> for a harmonic difference, using its zero boundary values and vanishing limit at infinity.
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