On the Hilbert space \(\widetilde {L}_{2}(\mathbb {T})\) the singular integral operator with two shifts and conjugation \(K=[aI+(a_{0}I+a_{1}U_{\alpha })U_{\beta }C]P_{+}+P_{-}\) is considered, where \(P_{\pm }\) are the Cauchy projectors, \(a,a_{0},a_{1}\) , are continuous functions on the unit circle \(\mathbb {T}\) , \(U_{\alpha }\) is a non-Carleman shift operator preserving the orientation on \(\mathbb {T}\) , \(U_{\beta }\) is a Carleman shift operator changing the orientation on \(\mathbb {T}\) , and C is the operator of complex conjugation. An estimate for the dimension of the kernel of the operator K is obtained.