= Solution
Substitute $U=R(r)F(\theta)$ into the axisymmetric <Laplace equation> and multiply by $r^2/(RF)$. The radial and angular terms must be separate constants:
$$
\frac1R\frac d{dr}(r^2R')=-\frac1{F\sin\theta}\frac d{d\theta}(\sin\theta F')=\lambda.
$$
Thus
$$
r^2R''+2rR'-\lambda R=0,\qquad
\frac1{\sin\theta}(\sin\theta F')'+\lambda F=0.
$$
With $x=\cos\theta$, the angular equation becomes
$$
(1-x^2)F_{xx}-2xF_x+\lambda F=0.
$$
Requiring boundedness on the entire polar interval, and using the supplied characterization of bounded <Legendre polynomial> solutions, gives $\lambda=n(n+1)$ and $F\propto P_n(\cos\theta)$ for $n=0,1,2,\ldots$. Trying $R=r^s$ gives $s(s+1)=n(n+1)$, hence $s=n$ or $s=-n-1$.
The globally angular-regular separated modes and their superposition are therefore
$$
\boxed{U_n=(A_nr^n+B_nr^{-n-1})P_n(\cos\theta),}
$$
$$
\boxed{U(r,\theta)=\sum_{n=0}^\infty(A_nr^n+B_nr^{-n-1})P_n(\cos\theta).}
$$
Coefficients and convergence are determined by the radial domain and boundary data. A domain containing the origin excludes negative radial powers for a regular solution. If the angular domain excludes a polar axis of the <spherical coordinate system> and singular angular behavior is allowed, additional Legendre-function branches are possible; the displayed family is the one regular over the complete sphere, as intended by the hint.
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