Restricted $K$- and $τ$-symmetries on multisoliton manifolds of the potential BKP and CKP equations

We investigate the reduction of generalized $K$-symmetries and $τ$-symmetries on the multisoliton manifolds of the potential BKP and CKP hierarchies. The $n$-soliton solution of BKP depends on wave-number parameters $k_m$, transverse parameters $l_m$, and phase parameters $ξ_{m0}$, defining a $3n$-dimensional manifold. Its restricted $K$-symmetries project onto an $n$-dimensional phase-shift subspace, while the restricted $τ$-symmetries engage the full $3n$-dimensional tangent space. This structure naturally yields a row-replacement determinant representation. In contrast, the $n$-soliton solution of CKP has no independent transverse parameters, reducing the manifold dimension to $2n$. The restricted $K$-symmetries again form an $n$-dimensional phase-shift representation, but the $τ$-symmetries now involve only $2n$ tangent directions. For CKP, the natural formulation treats elementary tangent symmetries as unknowns, producing a column-replacement determinant representation via Cramer's rule. The difference between these two determinant structures reflects not a difference in the underlying linear algebra, but rather the distinct parameterizations of the two soliton manifolds. Both formulas express explicitly how infinite-dimensional symmetry hierarchies collapse onto finite-dimensional multisoliton families.