On the Hilbert space the singular integral operator with non-Carleman shift and conjugation is considered, where are the Cauchy projectors, , , , are continuous functions on the unit circle , U is the shift operator and C is the operator of complex conjugation. We show how the symbolic computation capabilities of the computer algebra system Mathematica can be used to explore the dimension of the kernel of the operator K. The analytical algorithm [ADimKer-NonCarleman] is presented; several nontrivial examples are given.