The low-lying dipole strength distributions in the odd-mass isotopes 135,137 Ba were studied in nuclear resonance fluorescence experiments (NRF) performed at the Stuttgart Dynamitron facility using bremsstrahlung beams with end point energies of 4.1, 3.1, and 2.5 MeV. Numerous excited states, most of them unknown so far, were observed in the excitation energy range up to 4 MeV. Detailed spectroscopic information has been obtained on excitation energies, decay widths, decay branching ratios, and transition probabilities. The results for
137Ba are compared with calculations in the framework of the Quasiparticle-Phonon Model. The new data for 135,137 Ba complete the systematics of low-lying dipole excitations as observed for the even Ba isotopes 134,136,138Ba
in previous NRF experiments in Stuttgart. The complete systematics within the Ba isotopic chain, exhibiting a nuclear shape transition, is discussed with respect to E1two-phonon excitations, M1scissors mode excitations, and in regard to the new critical point symmetries.