It is very difficult (for me) to trace back the Kyburz ancestry. However, I did find out that Kyburz is a very common name in certain areas of Switzerland. They congregated around the area of Aaru which is along the Aaru River.
My great grandmother came from this area in Solothurn Canton.
Her name was Maria Kyburz, born about 1873 and immigrated to USA with 5 children around 1892.
There is a little town in California named Kyburz, supposedly after a the Duke of Kyburz. Don't know if this is fact or some local myth.