  In this work we derive the state of strain or stress under symmetry
conserving conditions in pseudomorphic lattices with monoclinic symmetry. We
compare surface vectors across the template epitaxial layer interface and
impose conditions of a stress free epitaxial layer. As a result, we demonstrate
the existence, in theory, of exactly three possible unit cells which can
establish onto a given template. We demonstrate this approach for a class of
templates with $(h0l)$ planes and $\beta$-(Al$_x$Ga$_{1-x}$)$_2$O$_3$ on
$(h0l)$ $\beta$-Ga$_2$O$_3$. We discuss the effects of composition $x$ and
surface orientation onto the formation of three elastically stable unit cells,
their strain and stress tensors, unit cell axes, unit cell volumes, lattice
spacing, elastic potential energies, and stress free directions. The previous
paradigm for epitaxial layer growth where the stress free direction is always
perpendicular to the growing surface is not generally valid for low symmetry
materials. In the example here, we find two possible competing domains with
stress free direction oblique to the surface of the template for almost all
planes $(h0l)$. We calculate the band-to-band transitions for
$\beta$-(Al$_{0.1}$Ga$_{0.9}$)$_2$O$_3$ on $(h0l)$ $\beta$-Ga$_2$O$_3$ using
the composition dependent deformation parameters and elastic coefficients
reported prevoiously [Korlacki~\textit{et al.} Phys. Rev. Appl.~\textbf{18},
064019 (2022)].
