Having recently hired an Austrian genealogist to research this Lackner family, I now know much more!
Anton, my grandfather, was born July 2, 1881 at Rappersdorf 10, Mühldorf Parish, Mölltal, Kärnten (Carinthia) Austria. He was illegitimate, and his mother was Lucia Lackner, born July 11, 1849 further up the Möll valley at Gastl, Teuchl Parish. Her father was Johann Aloÿs Lackner (1816-1870) and her mother was Luzia Rindler (b1808). Aloÿs's father was Vitus (Veit) Lackner (b1768), his father was also Veit (1723-1797), as was his father (married Christina Muesnig 1767). The family's stronghold was Stappitz, next to Mallnitz, which is up a valley just north of Obervellach. Obervellach is a few miles up the Möll valley from Rappersdorf.
I am now trying to see if this genealogist can find who Lucia may have married, and therefore find who Anton's half brothers and sisters were.
Anton is rumoured to have gone to the USA in his early teens, probably around 1894. He didn't arrive in NZ until about 1910. Who knows, he may have had a another family in the USA??? I wouldn't be surprised.