  By a classical result of Darboux, a foliation of a Riemannian surface has the
Graves property (also known as the strong evolution property) if and only if
the foliation comes from a Liouville net. A similar result of Blaschke says
that a pair of orthogonal foliations has the Ivory property if and only if they
form a Liouville net.
  Let us say that a geodesically convex curve on a Riemannian surface has the
Poritsky property if it can be parametrized in such a way that all of its
string diffeomorphisms are shifts with respect to this parameter. In 1950,
Poritsky has shown that the only closed plane curves with this property are
ellipses.
  In the present article we show that a curve on a Riemannian surface has the
Poritsky property if and only if it is a coordinate curve of a Liouville net.
We also recall Blaschke's derivation of the Liouville property from the Ivory
property and his proof of Weihnacht's theorem: the only Liouville nets in the
plane are nets of confocal conics and their degenerations.
  This suggests the following generalization of Birkhoff's conjecture: If an
interior neighborhood of a closed geodesically convex curve on a Riemannian
surface is foliated by billiard caustics, then the metric in the neighborhood
is Liouville, and the curve is one of the coordinate lines.
