The paper deals with a Hamiltonian, namely, with a semi-bounded self-adjoint operator
that is attributed to the problem of scattering of three one-dimensional particles with point interaction
in pairs, in other words, with δ-functional singular potential of interaction. The support of the potential
in the Hamiltonian coincides with a symmetric star-graph having six leads on the two-dimensional
plane. Due to the symmetry, we find that such a model is exactly solvable, which means that the
eigenfunctions of the discrete spectrum and the generalized eigenfunctions of the essential (absolutely
continuous) spectrum are determined explicitly, i.e., by quadrature. In this (first part) of our work we
describe the discrete spectrum and the eigenfunctions.