  We study the tunnelling trough a potential barrier of the system of two
quantum correlated particles. The system is considered in one dimension. The
interaction with the barrier and between particles is approximated by
$\delta$-potentials. Assuming that the particles have the same masses, we
reduce the problem to the set of independent planar scattering problems
corresponding to two-body flux symmetry with respect to the barrier. In order
to solve the problems we apply the method of Sommerfeld-Maluzhinets integral
transformation, which requires the solution of a set of functional equations.
We show that the sub-problem which is antisymmetric with respect to the center
of mass allows the solution in frames of the Bethe ansatz. So we give the exact
solution of this problem. For the subproblem which is symmetric we describe the
class of solutions for the functional equations and give an approach for
obtaining the solution.
