= Solution
Along the <circle> with increasing $\phi$, the <line element> is $d\boldsymbol r=a\sin\alpha\,\boldsymbol e_\phi d\phi$. Thus the two <circulations> are
$$
\oint_C\boldsymbol A^{(+)}\cdot d\boldsymbol r=2\pi\sin\alpha\tan(\alpha/2),\qquad
\oint_C\boldsymbol A^{(-)}\cdot d\boldsymbol r=-2\pi\sin\alpha\cot(\alpha/2).
$$
The half-angle identities simplify them to
$$
\boxed{\oint_C\boldsymbol A^{(+)}\cdot d\boldsymbol r=2\pi(1-\cos\alpha),\qquad
\oint_C\boldsymbol A^{(-)}\cdot d\boldsymbol r=-2\pi(1+\cos\alpha).}
$$
Here $0<\alpha<\pi$ so both potentials are defined on the whole curve.
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