  It was earlier conjectured by the second and the third authors that any
rational curve $g:{\mathbb C}P^1\to {\mathbb C}P^n$ such that the inverse
images of all its flattening points lie on the real line ${\mathbb R}P^1\subset
{\mathbb C}P^1$ is real algebraic up to a linear fractional transformation of
the image ${\mathbb C}P^n$. (By a flattening point $p$ on $g$ we mean a point
at which the Frenet $n$-frame $(g',g'',...,g^{(n)})$ is degenerate.) Below we
extend this conjecture to the case of meromorphic functions on real algebraic
curves of higher genera and settle it for meromorphic functions of degrees
$2,3$ and several other cases.
