In this paper we study realizations of infinite-dimensional Witt and Virasoro algebras. We obtain a complete description of realizations of the Witt algebra by Lie vector fields of first-order differential operators over the space ℝ3. We prove that none of them admits non-trivial central extension, which means that there are no realizations of the Virasoro algebra in ℝ3. We describe all inequivalent realizations of the direct sum of the Witt algebras by Lie vector fields over ℝ3. This result enables complete description of all possible (1+1)- dimensional partial differential equations that admit infinite dimensional symmetry algebras isomorphic to the direct sum of Witt algebras. In this way we have constructed a number of new classes of nonlinear partial differential equations admitting infinite-dimensional Witt algebras. So new integrable models which admit infinite symmetry algebra are obtained.

中文翻译：

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Witt 和 Virasoro 代数和可积方程的实现

在本文中，我们研究了无限维 Witt 和 Virasoro 代数的实现。我们通过空间 ℝ3 上的一阶微分算子的李向量场获得了对 Witt 代数实现的完整描述。我们证明它们都不承认非平凡中心扩展，这意味着ℝ3中没有Virasoro代数的实现。我们通过 ℝ3 上的李向量场描述了 Witt 代数的直和的所有不等价实现。该结果能够完整描述所有可能的 (1+1) 维偏微分方程，这些方程允许无限维对称代数同构为 Witt 代数的直接和。通过这种方式，我们构建了许多新的非线性偏微分方程类，允许无限维维特代数。