  Let $F/{\mathbb Q}_p$ be a finite unramified extension, let $k$ be a finite
extension of the residue field of $F$. We provide explicit constructions of
integral structures for all rank two \'{e}tale Lubin-Tate $(\varphi,{\mathcal
O}_F^{\times})$-modules over $k$. We construct algebraic families of such
integral structures and show that these comprehensively reflect the
degeneration behaviour of $(\varphi,{\mathcal O}_F^{\times})$-modules. These
results reveal new combinatorial structures of the moduli stack of
$(\varphi,{\mathcal O}_F^{\times})$-modules, and allow us, in particular, to
rederive the fact that the Serre weights assigned to a two dimensional ${\rm
Gal}(\overline{F}/F)$-representation over $k$ can be read off from the geometry
of the stack.
