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 strictly 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 strictly 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.

由达布克斯（Darboux）得出的经典结果是，当且仅当叶面来自Liouville网时，黎曼曲面的叶面才具有Graves属性（也称为强演化属性）。布拉施克（Blaschke）的类似结果说，一对且仅当它们形成Liouville网时，才具有象牙性质。

让我们说，如果可以通过参数化黎曼曲面上的严格测地线凸曲线来使其所有字符串微分形都相对于该参数发生位移，则它具有Poritsky属性。1950年，Poritsky证明了具有此属性的唯一闭合平面曲线是椭圆。

在本文中，我们证明，当且仅当它是Liouville网的坐标曲线时，黎曼曲面上的曲线才具有Poritsky属性。我们还记得布拉斯基（Blaschke）从科特迪瓦（Ivory）财产继承了利维尔（Liouville）财产，以及他对魏纳赫特定理的证明：飞机上唯一的利维尔网是共聚焦圆锥形及其退化的网。

这表明对Birkhoff猜想有以下一般化：如果台球焦散使黎曼面上的闭合严格测地线凸曲线的内部邻域成为台球焦散，则该邻域的度量为Liouville，并且该曲线是坐标线之一。