  Let M be a real analytic Riemannian manifold. An adapted complex structure on
$TM$ is a complex structure on a neighborhood of the zero section such that the
leaves of the Riemann foliation are complex submanifolds. This structure is
called entire if it may be extended to the whole of $TM$. We prove here that
the only real analytic Zoll metric on the $n$-sphere with an entire adapted
complex structure on $TM$ is the round sphere. Using similar ideas, we answer a
special case of an algebraization question raised by the first author,
characterizing some Stein manifolds as affine algebraic in terms of
plurisubharmonic exhaustion functions satisfying the homogeneous complex
Monge-Amp\`ere (HCMA) equation.
