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Given an affine rational complexity-one   $T$  -variety   $X$  , we construct an explicit embedding of   $X$   in affine space   ${\mathbb{A}}^n$  . We show that this embedding is well-poised, that is, every initial ideal of   $I_X$   is a prime ideal, and we determine the tropicalization   ${\mathrm{Trop}}(X^\circ)$  . We then study valuations of the coordinate ring   $R_X$   of   $X$   which respect the torus action, showing that for full rank valuations, the natural generators of   $R_X$   form a Khovanskii basis. This allows us to determine Newton–Okounkov bodies of rational projective complexity-one   $T$  -varieties, partially recovering (and generalizing) results of Petersen. We apply our results to describe all integral special fibres of   ${\mathbb{K}}^*\times T$  -equivariant degenerations of rational projective complexity-one   $T$  -varieties, generalizing a result of Süß and Ilten.

Rational Complexity-One $\boldsymbol{T}$ -Varieties Are Well-Poised