The essence of Stahl-Gonchar-Rakhmanov theory of symmetric contours as
applied to the multipoint Pad\'e approximants is the fact that given a germ of
an algebraic function and a sequence of rational interpolants with free poles
of the germ, if there exists a contour that is "symmetric" with respect to the
interpolation scheme, does not separate the plane, and in the complement of
which the germ has a single-valued continuation with non-identically zero jump
across the contour, then the interpolants converge to that continuation in
logarithmic capacity in the complement of the contour. The existence of such a
contour is not guaranteed. In this work we do construct a class of pairs
interpolation scheme/symmetric contour with the help of hyperelliptic Riemann
surfaces (following the ideas of Nuttall \& Singh and Baratchart \& the author.
We consider rational interpolants with free poles of Cauchy transforms of
non-vanishing complex densities on such contours under mild smoothness
assumptions on the density. We utilize ∂ˉ-extension of the
Riemann-Hilbert technique to obtain formulae of strong asymptotics for the
error of interpolation
