  The low-lying eigenstates of a one-dimensional (1D) system of many
impenetrable point bosons and one moving impurity particle with repulsive
zero-range impurity-boson interaction are found for all values of the
impurity-boson mass ratio and coupling constant. The moving entity is a
polaron-like composite object consisting of the impurity clothed by a co-moving
gray soliton. The special case with impurity-boson interaction of point
hard-core form and impurity-boson mass ratio $m_i/m$ unity is first solved
exactly as a special case of a previous Fermi-Bose (FB) mapping treatment of
soluble 1D Bose-Fermi mixture problems. Then a more general treatment is given
using second quantization for the bosons and the second-quantized form of the
FB mapping, eliminating the impurity degrees of freedom by a Lee-Low-Pines
canonical transformation. This yields the exact solution for arbitrary $m_i/m$
and impurity-boson interaction strength.
