= Solution
Set $U=R(r)\Theta(\theta)$ in the <Laplace equation>, multiply by $r^2/U$, and separate:
$$
\frac1R(r^2R')'=\lambda,\qquad \frac1{\sin\theta}(\sin\theta\Theta')'+\lambda\Theta=0.
$$
Putting $x=\cos\theta$ turns the angular equation into the <Legendre differential equation>. For a solution smooth on the whole spherical axis, the allowed eigenvalues are $\lambda=n(n+1)$ with angular functions $P_n(\cos\theta)$. The radial equation has the two solutions $r^n,r^{-n-1}$. Superposition yields the regular axisymmetric expansion
$$
\boxed{U(r,\theta)=\sum_{n=0}^\infty(A_nr^n+B_nr^{-n-1})P_n(\cos\theta).}
$$
The coefficients and convergence region are set by the domain and boundary data. Without regularity at the poles, singular Legendre functions and more general separated degrees would also be possible; they are excluded for this exterior spherical field.
The far-field matching fixes $A_1=v_0$ and all other growing coefficients to zero. The radial <Neumann boundary condition> at $r_0$ is
$$
\sum_{n=0}^\infty[nA_nr_0^{n-1}-(n+1)B_nr_0^{-n-2}]P_n(\cos\theta)=0.
$$
Independence of the angular modes gives $B_1=v_0r_0^3/2$ and all other decaying coefficients zero. Thus the <exterior harmonic potential with no flux through a sphere> is
$$
\boxed{U=v_0\left(r+\frac{r_0^3}{2r^2}\right)\cos\theta.}
$$
Its derivative vanishes on the sphere and its difference from $v_0r\cos\theta$ tends to zero at infinity. A spatial constant may be added if the far-field condition is intended only to specify the gradient or the leading growing term; matching the potential difference to zero fixes that constant to zero.
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