A characterization of the Kadomtsev–Petviashvili hierarchy of type C (CKP) in terms of the KP tau-function is given. Namely, we prove that the CKP hierarchy can be identified with the restriction of odd times flows of the KP hierarchy on the locus of turning points of the second flow. The notion of CKP tau-function is clarified and connected with the KP tau function. Algebraic–geometrical solutions and in particular elliptic solutions are discussed in detail. A new identity for theta-functions of curves with holomorphic involution having fixed points is obtained.