Eigenfunctions are shown to constitute privileged coordinates of self-dual Einstein spaces with the underlying governing equation being revealed as the general heavenly equation. The formalism developed here may be used to link algorithmically a variety of known heavenly equations. In particular, the classical connection between Plebanski's first and second heavenly equations is retrieved and interpreted in terms of eigenfunctions. In addition, connections with travelling wave reductions of the recently introduced TED equation which constitutes a 4+4-dimensional integrable generalisation of the general heavenly equation are found. These are obtained by means of (partial) Legendre transformations. As a particular application, we prove that a large class of self-dual Einstein spaces governed by a compatible system of dispersionless Hirota equations is genuinely four-dimensional in that the (generic) metrics do not admit any (proper or non-proper) conformal Killing vectors. This generalises the known link between a particular class of self-dual Einstein spaces and the dispersionless Hirota equation encoding three-dimensional Einstein-Weyl geometries.

中文翻译：

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自对偶爱因斯坦空间和一般天堂方程。作为坐标的特征函数

特征函数被证明构成了自对偶爱因斯坦空间的特权坐标，其中潜在的控制方程被揭示为一般的天堂方程。这里开发的形式主义可用于在算法上链接各种已知的天堂方程。特别是，Plebanski 的第一个和第二个天堂方程之间的经典联系是根据本征函数来检索和解释的。此外，还发现了与最近引入的 TED 方程的行波减少的联系，该方程构成了一般天堂方程的 4+4 维可积推广。这些是通过（部分）勒让德变换获得的。作为一个特殊的应用，我们证明了由无色散 Hirota 方程的兼容系统控制的一大类自对偶爱因斯坦空间是真正的四维空间，因为（通用）度量不允许任何（适当的或非适当的）共形杀伤向量。这概括了特定类别的自对偶爱因斯坦空间与编码三维爱因斯坦-外尔几何的无色散 Hirota 方程之间的已知联系。