Her home, at times, was a shaded bench with a sight line toward the White House, in a tiny park used regularly by employees of the nearby World Bank.

Jiamei Tian often sat alone, keeping a close eye on her cart and the blue laundry bag that held her belongings. Off in the distance, over her left shoulder, stood the Lincoln Memorial, and behind her stretched the Mall. She occupied a park managed by the National Park Service, the same agency that protects some of the national treasures that Tian is suspected of defacing with splotches of mint green paint.

Why Tian might have tarnished some of the most visible emblems of the capital city remains a mystery. Those landmarks surrounded her uncertain existence, which has proved difficult to trace.

Tian is from China, doesn’t speak much English, and is believed by authorities to suffer from mental illness. But little else has emerged about this 58-year-old woman since her arrest last week after a four-day outbreak of vandalism in which churches, statues and the nation’s hallowed memorial to Abraham Lincoln were splashed with paint.

A motive for the attacks remains as elusive as Tian’s story, both in China and in the United States. Only fragments have emerged from police reports, court documents, hearings and interviews with investigators, advocates for the homeless, church administrators, and the men and women who live in the park.

Tian’s given name means “beautiful woman” in Chinese. The regulars in Edward R. Murrow Park, at 1800 Pennsylvania Ave. NW, called her “Jan.” She has been spotted at churches, soup kitchens and green spaces from downtown Washington to the U.S. Capitol, at Thomas Circle and as far north as the Chevy Chase traffic circle.

Some of Tian’s travels are documented in police reports.

Now in custody, she has been charged with spattering green paint on religious statues and an organ at Washington National Cathedral, where officers found her hiding in a restroom stall last Monday with green paint on her shoes.

Authorities said Tian is suspected of similar attacks at Luther Place Memorial Church at Thomas Circle in Northwest Washington — where paint mixed with feces and urine was found on the choir organ — and of vandalizing statues outside the church and the Smithsonian Castle. They also linked her to paint found at the Lincoln Memorial. A detective testified in D.C. Superior Court on Friday that paint footprints under the inscription of the Gettsyburg Address match Tian’s shoes.

Tian has declined to tell investigators where she lived in the District, where she is from or how she got here, authorities said. Federal immigration officials would say only that she came to the United States on a short-term visa that expired July 27, in the midst of the rash of vandalism. Citing privacy concerns, they declined to reveal the date she arrived in this country or her city of entry. One of Tian’s attorneys confirmed in court last week that she came to this country on a tourist visa.

When she was arrested, Tian was carrying a 12-ounce bottle of cocktail sauce, a clear bag, a tie-dye shirt, shorts and some multicolored socks, according to a police report.

She has said little to anyone. When first confronted by a police officer in the cathedral restroom, she walked away and placed a can of green paint she was holding in a yellow bag, according to a police report.

But Tian did appear to display some emotion at the hearing Friday in Superior Court, where she followed the proceedings with help from a Mandarin interpreter. After the judge ordered her released to a halfway house, Tian leaned over to her attorney, smiled and mouthed, “Thank you.”

Police said Tian told them that she had been in Los Angeles, and a federal official said authorities believe that is where she entered the country. A spokesman for U.S. Customs and Border Protection would not comment.

A social worker who met Tian in Murrow Park said Tian told her that she had been in Arlington County. The judge at Tian’s arraignment said she had been in the District for only two days before her arrest, but advocates for the homeless said they saw her in the Foggy Bottom park in winter and spring.

The administrator at Luther Park Memorial Church said Tian was known to congregants and attended Sunday services the day paint was strewn in the choir loft. Advocates for the homeless said she had been seen at various shelters, soup kitchens and churches, but administrators of those organizations refused to confirm those reports, citing privacy concerns.

Tian’s name is common in China, and little is known of her life there. Her home province has not been made public, nor have many details been published about her in the Chinese media. The Chinese Embassy in Washington did not respond to requests for interviews.

If she has had other encounters with law enforcement agencies, they weren’t reported. Officials with D.C. police, U.S. Park Police and Smithsonian security said they have no records of interactions with Tian, arrests or otherwise. Based on her behavior since her arrest, D.C. Police Chief Cathy L. Lanier said Tian may have “mental-health issues.”

Interviews with a dozen or more homeless men and women in Murrow Park revealed little about her. One man, who identified himself only as “Stretch,” said he had seen her making the rounds of downtown parks. Stretch said he was 55 and had been homeless for 15 years.

“I’ve seen her a few times,” he said after studying a photo of Tian taken during her arrest. He knew her as “Jan” and said she kept to herself, preferring to sit alone on her bench. He never talked with her.

The park is named for Edward R. Murrow, a famous journalist known, in part, for his radio broadcasts during World War II, and it is one of several Park Service sites in the District related to the war.

The park is bisected by Pennsylvania Avenue, and each triangular section has a shaded path lined with benches and surrounded by bushes, in which the homeless store their belongings. Trash bags filled with blankets and clothes, books, coolers and food were visible on a recent weekday. The men and women who call the park home said most of them live on one side of Pennsylvania Avenue or the other. They said Tian stayed on the north side.

A man who works at the World Bank, which towers over Murrow Park from its location on H Street NW, said that bank employees help maintain the park but that he had not seen Tian there.

The social worker — who works for a homeless advocacy group and spoke on the condition that she not be named because she was discussing a client — said she had encountered Tian in the park several times since winter. Tian would not give the social worker her name.

She liked to keep herself and her clothes clean; she asked, in broken English, for laundry detergent and toiletries when approached by an advocate for the homeless. She wanted everything sealed “or she wouldn’t take them,” the advocate said. She dressed in layers and preferred multicolored socks and canvas shoes with rubber soles.

“She would not discuss anything with me,” the advocate said. “She wouldn’t show her ID. She doesn’t tell anyone where she goes. All she revealed is that she had been in Virginia and she wanted stuff to do her laundry with.”

Tian sat on a bench facing north, apparently to give her the best view of the park — and to avoid people she saw approaching, the social worker and Stretch said.

“Jail is not going to help her,” the social worker said. “She has family somewhere.”

Keith L. Alexander, Trishula Patel, Lena H. Sun and William Wan in Beijing contributed to this report.