The neutron-rich nuclei $^{109}$Pd and $^{111}$Pd were produced as fission fragments following the $^{30}$Si + $^{168}$Er reaction at 142 MeV. Using the identification based on the coincidences with the complementary fission fragments, the only positive-parity bands observed so far in $^{109}$Pd and $^{111}$Pd emerged from this work. A band, built on top of the 5/2$^+$ ground state exhibiting $Delta I$ = 1 energy-level staggering, was observed in each of these nuclei. Both nuclei of interest, $^{109}$Pd and $^{111}$Pd, are suggested to lie in the transitional region of Pd isotopes of maximum $gamma$-softness. The ground states of both nuclei are predicted by TRS calculations to be extremely $gamma$-soft with shallow triaxial minima. The first crossing in the new bands is proposed to be due to an alignment of $h^2_{11/2}$ neutrons.
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