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In this paper we describe the orbit structure of  $C^2$-actions of  ${ℝ}^2$  on the solid torus  $S^1\times D^2$  having  $S^1\times \left\{0\right\}$  and  $S^1\times \partial D^2$  as the only compact orbits, and  $S^1\times \left\{0\right\}$  as singular set.