  We consider two classes of non-toric log del Pezzo $\mathbb{C}^*$-surfaces:
on the one side the 1/3-log canonical ones and on the other side those of
Picard number one and Gorenstein index at most 65. In each of the two classes
we figure out the surfaces admitting a K\"ahler-Einstein metric, a
K\"ahler-Ricci soliton and those allowing a Sasaki-Einstein metric on the link
of their anticanonical cone. We encounter examples that admit a
K\"{a}hler-Ricci soliton but no Sasaki-Einstein cone link metric.
