  A particle-core Hamiltonian is used to describe the lowest parity partner
bands $K^{\pi}=1/2^{\pm}$ in
  $^{219}$Ra, $^{237}$U and $^{239}$Pu, and three parity partner bands,
$K^{\pi}=1/2^{\pm}, 3/2^{\pm}, 5/2^{\pm}$, in $^{227}$Ra. The core is described
by a quadrupole and octupole boson Hamiltonian which was previously used for
the description of four positive and four negative parity bands in the
neighboring even-even isotopes. The particle-core Hamiltonian consists of four
terms: a quadrupole-quadrupole, an octupole-octupole, a spin-spin and a
rotational $\hat{I}^2$ interaction, with $\hat {I}$ denoting the total angular
momentum. The single particle space for the odd nucleon consists of three
spherical shell model states, two of positive and one of negative parity. The
product of these states with a collective deformed ground state and the
intrinsic gamma band state generate, through angular momentum projection, the
bands with $K^{\pi}=1/2^{\pm},3/2^{\pm},5/2^{\pm}$, respectively. In the space
of projected states one calculates the energies of the considered bands. The
resulting excitation energies are compared with the corresponding experimental
data as well as with those obtained with other approaches. Also, we searched
for some signatures for a static octupole deformation in the considered odd
isotopes. The calculated branching ratios in $^{219}$Ra agree quite well with
the corresponding experimental data.
