On a matrix constrained CKP hierarchy

The algebraic structures of integrable hierarchies play an important role in the study of soliton equations. In this paper, we use splitting theory to give a matrix representation of a constrained CKP hierarchy, which can be considered as a generalization of the $\hat{A}_{2n}^{(2)}$-KdV hierarchy and the constrained KP hierarchy. An equivalent construction in terms of the pseudo-differential operator is discussed. Darboux transformations, scaling transformation and tau functions $\ln \tau_f$ for this constrained hierarchy are studied. Moreover, we present formulas for the Virasoro vector fields on $\ln \tau_f$ for the $\hat{A}_{2 n}^{(2)}$-KdV hierarchy.