A small shoe found four blocks from where a Washington state woman said she left her 2-year-old son does not belong to the missing boy, police said.

A couple walking in Bellevue, Wash., found a baby shoe very near where Julia Biryukova said she left her son, Sky Metalwala, on Sunday before he disappeared. 

Police looking for clues in the boy's disappearance examined the shoe as possible evidence but determined Thursday night that it did not belong to the toddler.

Biryukova alerted police Sunday morning after she said she discovered Sky missing from her car in Bellevue.

Biryukova told police that she was driving her sick son to a hospital when she ran out of gas. She claimed she left the boy buckled into his car seat, sleeping, while she and her 4-year-old daughter walked to a Chevron gas station, about a mile away. When the two returned to the unlocked car, the child was gone, according to Biryukova.

But investigators said Biryukova's story "doesn't add up," and is very similar to an episode of "Law & Order: SVU," which aired a day before the child disappeared.

On Thursday, authorities appealed to Biryukova to come forward and provide more information about the boy's disappearance.   

"I want to make a public invitation to Julia, through her attorney, that we are open and willing to interview her and take information whenever she's ready," Bellevue Maj. Mike Johnson said at a news conference on Thursday.

Biryukova has so far refused to take a lie detector test, according to authorities. The child's father, Solomon Metalwala, is expected to take a second polygraph test after the first one was ruled inconclusive.

Biryukova and Metalwala are in the midst of a bitter divorce battle.

In court documents, Metalwala said his wife suffers from mental illness and had dreams about killing the couple's children, according to Fox affiliate KCPQ-TV. 

Biryukova has accused Metalwala of domestic violence, claiming he assaulted her in front of the children in December 2009.

Police have said they suspect foul play in the disappearance of Sky, who is of Indian and Caucasian descent with brown eyes and dark hair. He was last seen wearing a dark green hooded sweatshirt and blue and gray-striped pants, according to his mother.

Authorities have said they found no gas can in their investigation. The also found that Biryukova's car had plenty of gas in it when its fuel tank was checked. 

"There was a sufficient amount of gas in that vehicle to run it for a considerable distance," Johnson said. 

Biryukova had previously been cited for leaving a child unattended in a vehicle. In 2009, Redmond police cited Biryukova and Metalwala for leaving a child unattended in a vehicle. The case was dismissed after they agreed to probation, community service and a 10-week parenting class.

Fox News' Dan Springer contributed to this report.