  Two-leg spin-$\frac12$ ladders with anisotropy and two different dimerization
patterns are analyzed at zero temperature. This model is equivalent to a
modulated interacting (Kitaev) ladder. The Hartree-Fock mean-field
approximation reduces the model to a sum of two quadratic effective Majorana
Hamiltonians, which are dual to a sum of two (even/odd) XY quantum chains in
the alternating transverse fields. The mapping between the effective
Hamiltonian of the ladder and the pair of the dual XY chains considerably
simplifies calculations the order parameters and analysis of the hidden
symmetry breaking. The ground-state phase diagram of the staggered ladder
contains nine phases, four of them are conventional antiferromagnets, while the
other five possess non-local brane orders. Using the dualities and the newly
found exact results for the local and string order parameters of the transverse
XY chains, we were able to find analytically all the magnetizations and the
brane order parameters for the staggered case, as functions of the renormalized
couplings of the effective Hamiltonian. The columnar ladder has three
ground-state phases and does not possess magnetic long-ranged order. The brane
order parameters for these three phases are calculated numerically from the
Toeplitz determinants. We expect this study to motivate the search for the real
spin-Peierls anisotropic ladder compounds which can undergo the predicted
quantum phase transitions with gap closures and distinct brane orders.
