  We use scanning tunneling microscopy (STM) to study charge density wave (CDW)
states in the rare-earth di-telluride, CeTe$_{2}$. In contrast to previous
experimental and first-principles studies of the rare-earth di-tellurides, our
STM measurements surprisingly detect a unidirectional CDW with $\textit{q}$ ~
0.28 $\textit{a}$*, which is very close to what is found in experimental
measurements of the related rare-earth tri-tellurides. Furthermore, in the
vicinity of an extended sub-surface defect, we find spatially-separated as well
as spatially-coexisting unidirectional CDWs at the surface of CeTe$_{2}$. We
quantify the nanoscale strain and its variations induced by this defect, and
establish a correlation between local lattice strain and the
locally-established CDW states. Our measurements probe the fundamental
properties of a weakly-bound two-dimensional Te-sheet, which experimental and
theoretical work has previously established as the fundamental component
driving much of the essential physics in both the rare-earth di- and
tri-telluride compounds.
