In this paper, we mainly study the additional symmetry and $\tau$ functions of a multi-component Gelfand-Dickey hierarchy which includes many classical integrable systems, such as the multi-component KdV hierarchy and the multi-component Boussinesq hierarchy. With other kinds of reductions, we can derive a B type multi-component Gelfand-Dickey hierarchy and a C type multi-component Gelfand-Dickey hierarchy. In our research, the additional flows of the additional symmetries can not all survive. By calculating, we find that the generator of the additional symmetry of the C type multi-component Gelfand-Dickey hierarchy is different from that of the B type multi-component Gelfand-Dickey hierarchy, while the forms of their surviving additional flows are the same.

中文翻译：

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多分量 Gelfand-Dickey 系统的 Virasoro 对称性

在本文中，我们主要研究多分量 Gelfand-Dickey 层次结构的附加对称性和 $\tau$ 函数，该层次包括许多经典的可积系统，例如多分量 KdV 层次结构和多分量 Boussinesq 层次结构。通过其他类型的归约，我们可以推导出 B 型多分量 Gelfand-Dickey 层次结构和 C 型多分量 Gelfand-Dickey 层次结构。在我们的研究中，附加对称性的附加流不可能全部存在。通过计算，我们发现C型多分量Ge​​lfand-Dickey层次结构的附加对称性的生成器与B型多分量Ge​​lfand-Dickey层次结构的附加对称性生成器不同，但它们的幸存附加流的形式相同.