Special solutions to five autonomous integrable partial difference equations via the third and sixth Painlevé equations and the Garnier system in two variables

In this paper, we study special solutions of five autonomous integrable partial difference equations (P$\mathrm{\Delta }$Es). More precisely, we show that these P$\mathrm{\Delta }$Es admit special solutions that are described by non-autonomous ordinary difference equations arising from Bäcklund transformations of the third and sixth Painlevé equations and the Garnier system in two variables. This result provides a new perspective on the relationship between autonomous integrable P$\mathrm{\Delta }$Es and Painlevé-type dynamics.