Though police maintain that the mother of missing 2-year-old Sky Metalwala has cooperated with the investigation, holes are beginning to appear in her story.

Police said Wednesday they were able to determine that Julia Biryukova’s silver Acura Integra – which she said ran out of gas Sunday morning – had enough fuel to drive for a considerable amount of time. Combined with her unwillingness to take a lie-detector test, Biryukova has become a source of police frustration.

“The story doesn’t add up, the whereabouts of Sky are unknown, and Mom’s story is falling apart day to day; and her lack of cooperation so far in providing more information proactively and submitting to a polygraph doesn’t help,” said Bellevue police Maj. Mike Johnson.

City crews are continuing to test the car’s mechanics Thursday to determine if anything is wrong with the vehicle.

Johnson said police will discuss the possibility of considering Biryukova a person of interest in the case, but she has answered all police questions through her attorney, and has not been named a suspect.

The focus of the investigation continues to be getting information from those closest to Sky and Biryukova, who lives in Redmond, so they can piece together the days and weeks that lead up to the boy being reported as missing Sunday.

“The people that were close to Sky and Julia are the people that know what happened to Sky,” Johnson told reporters Wednesday morning. “If everyone cooperates, we should have answers we need.”

The boy’s father, Solomon Metalwala will take a second lie-detector test after the first one proved to be too emotionally trying.

Solomon and Biryukova were going through a divorce and a custody battle before the boy disappeared.

A second individual, whom Johnson called a “close friend” has come in for polygraph test, police said.

Experts have interviewed Sky’s 4-year-old sister, a talk Johnson called “frustrating.” He said they received very few answers to their questions, but the daughter did confirm that Sky was in the car that morning wrapped in a blanket.

Police noted that Biryukova had numerous pictures of the 4-year-old girl on her Facebook page and in the home, but no pictures of the boy.

Sky was reported missing Sunday morning when he was allegedly left in the car, with at least one door unlocked, while Biryukova and her 4-year-old daughter walked to a nearby gas station. Biryukova told police she was taking Sky to Overlake Hospital Medical Center because he was dealing with an undisclosed illness.

Other than polygraph tests, Johnson said resources from the FBI, Bellevue, Redmond and King County will focus on DNA tests from samples found in Biryukova’s Acura Integra and her Redmond apartment.

Johnson said police will conduct DNA tests of interview subjects and the boy’s family to see if they are left with any suspect samples.

Police and FBI agents searched Biryukova’s downtown Redmond apartment and questioned neighbors and only one neighbor reported seeing the boy within the last two weeks. But Johnson said this wasn’t an unusual circumstance because the family is known to be reclusive.

Johnson said he was “incredulous” of Biryukova’s decision and movements on the morning when she left Sky in the unlocked car near the 2400 block of 112th Avenue Northeast.

“Many of us are parents as well as police officers, and when you hear this story that mom has told, you have to scratch your head and wonder why,” Johnson said. “If that’s what really what happened, why? This kid was sick; she was on the way to the hospital with him. Beyond that, why would you leave a 2-year-old alone in the car?”

If Sky was indeed in the vehicle that day, it wouldn’t be the first time he was left alone in a car.

Solomon and Biryukova were previously charged with reckless endangerment and leaving a child unattended in a sports utility vehicle in 2009 in a Target parking lot in Redmond when Sky was three months old, according to court documents. Police had to ask that the vehicle’s owners be paged in order to get the parents to come out to the vehicle, the documents said.

The charges were dismissed in February of this year after the parents took court-ordered parenting classes and completed community service, the documents stated.

Police along with resources from King County combed the nearby land looking for the boy, but the search came up empty, and it was suspended Sunday night.

The boy’s father Solomon lives in Kirkland with his younger brother, Sam, who said he believes someone kidnapped his nephew.

“Somebody has the child,” he said. “We just don’t know who it is.”

The boy’s disappearance comes at the height of a custody fight between Biryukova and Solomon, who is puzzled by Biryukova’s account of the details leading up to his son’s disappearance.

“I don’t understand why she would leave a little baby in the car,” Solomon told reporters at his Bellevue attorney’s office Monday afternoon. “The whole story is puzzling.”

Sam said Biryukova has not allowed Solomon to see either of his children since last November.

Biryukova claimed in court documents that Solomon had an anger problem and was abusive, while Solomon said in other court documents that Biryukova was battling a mental illness and was a threat to her children.

Sam confirmed that the couple agreed to a tentative compromise just last week that would allow Solomon to have some visitation with Sky and his older sister.

Solomon filed for divorce in June 2010 and that same month Biryukova requested a protection order and in it she said her husband had been physically abusive toward her and had threatened to kill her if she sought custody of their children.

Last December, a social worker determined there was “preponderance of evidence” that Solomon had struck his daughter.

Meanwhile, Solomon alleged Biryukova was battling a mental illness, was suicidal and had dreams of harming her children, according to a divorce court document filed in King County Superior Court.”We’re trying to think very positive,” said Sam, Solomon’s brother.

“We just hope that he’s safe and someone will bring him back.”