  We study the low-energy properties and, in particular, the magnetization
process of a spin-1/2 Heisenberg $J_1-J_2$ sawtooth and frustrated chain (also
known as zig-zag ladder) with a spatially anisotropic $g$-factor. We treat the
problem both analytically and numerically while keeping the $J_2/J_1$ ratio
generic. Numerically, we use complete and Lanczos diagonalization as well as
the infinite time-evolving block decimation (iTEBD) method. Analytically we
employ (non-)Abelian bosonization. Additionally for the sawtooth chain, we
provide an analytical description in terms of flat bands and localized magnons.
By considering a specific pattern for the $g$-factor anisotropy for both
models, we show that a small anisotropy significantly enhances a magnetization
plateau at half saturation. For the magnetization of the frustrated chain, we
show the destruction of the $1/3$ of the full saturation plateau in favor of
the creation of a plateau at half-saturation. For large anisotropies, the
existence of an additional plateau at zero magnetization is possible. Here and
at higher magnetic fields, the system is locked in the half-saturation plateau,
never reaching full saturation.
