  We calculate the odd-parity, radiative ($\ell \ge 2$) parts of the metric
perturbation in Lorenz gauge caused by a small compact object in eccentric
orbit about a Schwarzschild black hole. The Lorenz gauge solution is found via
gauge transformation from a corresponding one in Regge-Wheeler gauge. Like the
Regge-Wheeler gauge solution itself, the gauge generator is computed in the
frequency domain and transferred to the time domain. The wave equation for the
gauge generator has a source with a compact, moving delta-function term and a
discontinuous non-compact term. The former term allows the method of extended
homogeneous solutions to be applied (which circumvents the Gibbs phenomenon).
The latter has required the development of new means to use frequency domain
methods and yet be able to transfer to the time domain while avoiding Gibbs
problems. Two new methods are developed to achieve this: a partial annihilator
method and a method of extended particular solutions. We detail these methods
and show their application in calculating the odd-parity gauge generator and
Lorenz gauge metric perturbations. A subsequent paper will apply these methods
to the harder task of computing the even-parity parts of the gauge generator.
