On the Hilbert space (L) over tilde (2)(T) the singular integral operator with non-Carleman shift and conjugation K = P+ +(aI + AC)P- is considered, where P-+/- are the Cauchy projectors, A = (m)Sigma(j=0) a(j)U(j), a, a(j), j = (1, m) over bar, are continuous functions on the unit circle T, U is the shift operator and C is the operator of complex conjugation. Some estimates for the dimension of the kernel of the operator K are obtained.