Conjugacy word problem in the tree product of free groups with a cyclic amalgamation

The conjugacy word problem in the tree product of free groups with a cyclic amalgamation is solved in the positive. This result generalizes the known result obtained by S. Lipschutz for the free product of two free groups with cyclic amalgamation. Solution of the main problem involves proving the solvability of the problem of intersection of a finitely generated subgroup of a given class of groups with a cyclic subgroup belonging to the factor of the main group; the solvability of the problem of intersection of a coset of a finitely generated subgroup with a cyclic subgroup belonging to a free factor.