Police are continuing to appeal for sightings since Wednesday of a white Hyundai vehicle used by Lena Zhang Harrap's alleged killer.
Its registration is EGZ962 and it has distinctive brown tape around the passenger side wing mirror and back window.
Harrap was a regular walker but never made it back from an early morning walk at Ōwairaka Park, in Mt Albert, on Wednesday.
Her alleged killer, a 31-year-old man, appeared in the Auckland District Court on Friday, where has been remanded without plea to appear in the Auckland High Court next month.
Detective Inspector Aaron Pascoe said investigators would complete their scene examination on Saturday.
He said information from the public continued to help investigators piece together what had occurred.
"We again extend our condolences to Lena’s family and reiterate their pleas for privacy at this difficult time."
Zandra Vaccarino of the New Zealand Down Syndrome Association encouraged parents with children with Down syndrome not to let Harrap's death impact their independence.
"Lena’s death has shocked our community to the core and many parents have expressed increased fear and anxiety about giving their children with Down syndrome the independence any person deserves.
"We totally understand this sentiment, but we want to encourage all families not to give into this fear by keeping their loved ones at home.
"The reality is that this horrific incidence is an anomaly in New Zealand and we know that a person with a learning disability is safer in an inclusive community where they can fully participate in all aspects of life."
Vaccarino said Harrap was a "wonderful example" of someone fully immersed in her local community.
"This crime has sent shock waves through New Zealand as we grapple with the fact that a perpetrator targeted Lena because of an imbalance in power.
"We hope that this crime will galvanise a collective response to take action to put safeguards in place in our communities so that disabled people can continue to be fully included in our community," she said.
"We know that disabled people are safer when they are participating in their community and not isolated as highlighted by the current work of the Royal Commission into Abuse that has evidenced the extensive level of institutionalised abuse.
"Leading a full independent life is a human right for disabled people and inclusive communities are safer and healthier communities."
Anyone with information relating to the case, or sightings of the vehicle, is asked to call 105, quoting file number 210922/0539.