In this paper, under the constraints of the BKP(CKP) hierarchy, a crucialobservation is that the odd dynamical variable $u_{2k+1}$ can be explicitlyexpressed by the even dynamical variable $u_{2k}$ in the Lax operator $L$through a new operator $B$. Using operator $B$, the essential differencesbetween the BKP hierarchy and the CKP hierarchy are given by the flow equationsand the recursion operators under the $(2n+1)$-reduction. The formal formulasof the recursion operators for the BKP and CKP hierarchy under$(2n+1)$-reduction are given. To illustrate this method, the two recursionoperators are constructed explicitly for the 3-reduction of the BKP and CKPhierarchies. The $t_7$ flows of $u_2$ are generated from $t_1$ flows by theabove recursion operators, which are consistent with the corresponding flowsgenerated by the flow equations under 3-reduction.
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机译：在本文中，在BKP（CKP）层次结构的约束下，至关重要的观察是，奇数动态变量$ u_ {2k + 1} $可以由Lax运算符$中的偶数动态变量$ u_ {2k} $明确表示。通过新的运算符$ B $ L $。使用$ B $运算符，BKP层次结构和CKP层次结构之间的本质差异由流方程和$（2n + 1）$归约下的递归运算符给出。给出了$（2n + 1）$-归约下BKP和CKP层次结构递归运算符的形式公式。为了说明此方法，显式构造了两个递归运算符，用于BKP和CKPhierarchies的3-reduction。 $ u_2 $的$ t_7 $流量是由上述递归运算符从$ t_1 $的流量生成的，与3归约下由流量方程生成的相应流量一致。
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