They moved to New Zealand after Graham Dickason accepted a job at Timaru Hospital.
He is an orthopaedic surgeon.
It is understood he had been at a work function with colleagues and arrived home to find the grisly scene just two weeks after leaving managed isolation.
Details of the alleged murder have been suppressed.
Graham Dickason has shared images on Facebook of him in South Africa and it appears he is spending time camping and in the outdoors.
A source said he would not be back in New Zealand for his wife's next court appearance on December 17.
No return date has been set, the Herald has learned.
Details of any funeral of farewell for the little girls have not been shared by the family.
After Graham Dickason found his children a neighbour found him extremely distressed and called 111.
Lauren Dickason was unresponsive and in a critical condition when her husband arrived home.
She was taken to Timaru Hospital for treatment and was stable enough by Friday night to speak with police.
She was charged with murder soon after.
In September a service was held in Pretoria in memory of Liané, Maya and Karla.
Their uncle Graham Fawkes did not want to speak in-depth about what had happened to his nieces but shared photos with the Herald that were taken at a "celebration" of the girls earlier in the day.
"We love them and we will always love them," Fawkes told the Herald.
"You were so creative and loved your stickers and sparkles.
"You were kind and always cared deeply for those around you.
"You were a loving sister with a soft heart."
Karla was born with a cleft lip and her grandparents said last week she did not have the best start in life - but she overcame that and was brave and an explorer.
At the service she was described as "a beautiful little girl who was feisty, independent and strong".
If you are worried about your or someone else's mental health, the best place to get help is your GP or local mental health provider. However, if you or someone else is in danger or endangering others, call police immediately on 111.
OR IF YOU NEED TO TALK TO SOMEONE ELSE:
• LIFELINE: 0800 543 354 or 09 5222 999 within Auckland (available 24/7) • SUICIDE CRISIS HELPLINE: 0508 828 865 (0508 TAUTOKO) (available 24/7) • YOUTHLINE: 0800 376 633 ,free text 234 or email talk@youthline.co.nz or online chat. • NEED TO TALK? Free call or text 1737 (available 24/7) • KIDSLINE: 0800 543 754 (available 24/7) • WHATSUP: 0800 942 8787 (1pm to 11pm) • DEPRESSION HELPLINE: 0800 111 757 • SAMARITANS – 0800 726 666.