As a generalization of the integrable extended Toda hierarchy and a reduction of the extended multicomponent Toda hierarchy, from the point of a commutative subalgebra of $gl(2,\mathbb{C})$, we construct a strongly coupled extended Toda hierarchy(SCETH) which will be proved to possess a Virasoro type additional symmetry by acting on its tau-function. Further we give the multi-fold Darboux transformations of the strongly coupled extended Toda hierarchy.

中文翻译：

---

强耦合扩展 Toda 层次结构及其 Virasoro 对称性

作为可积扩展 Toda 层次结构的推广和扩展多分量 Toda 层次结构的简化，从 $gl(2,\mathbb{C})$ 的交换子代数的角度，我们构造了一个强耦合扩展 Toda 层次结构（SCETH ) 将通过作用于其 tau 函数被证明具有 Virasoro 型附加对称性。此外，我们给出了强耦合扩展 Toda 层次结构的多重 Darboux 变换。