did she have a relationship with the first singer?
Yes - In November 2002, Spears's relationship with Justin Timberlake ended after three years.
Britney Spears
In June 2002, Spears opened her first restaurant, Nyla, in New York City, but terminated her relationship in November, citing mismanagement and "management's failure to keep her fully apprised". In July 2002, Spears announced she would take a six-month break from her career; however, she went back into the studio in November to record her new album. Spears's relationship with Justin Timberlake ended after three years. In December 2002, Timberlake released the song "Cry Me a River" as the second single from his solo debut album. The music video featured a Spears look-alike and fueled the rumors that she had been unfaithful to him. As a response, Spears wrote the ballad "Everytime" with her backing vocalist and friend Annet Artani. 
cry me a river justin timberlake britney spears "Cry Me a River" is a song by Justin Timberlake. The lyrics were inspired by Timberlake's former relationship with singer Britney Spears. who produced it? Timbaland who is the first singer mentioned above? American singer, songwriter, actor, and record producer. who is the second singer mentioned? American singer, songwriter, dancer, and actress. what were a few of her greatest hit songs? Baby One More Time;  Oops!... I Did It Again ; Toxic
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Britney Spears [SEP] 2001–2002: "Britney" and "Crossroads"
In February 2001, Spears signed a $7–8 million promotional deal with Pepsi, and released another book co-written with her mother, titled "A Mother's Gift". Her self-titled third studio album, "Britney", was released in November 2001. While on tour, she felt inspired by hip hop artists such as Jay-Z and The Neptunes and wanted to create a record with a funkier sound. The album debuted at number one in the "Billboard" 200 and reached top five positions in Australia, the United Kingdom and mainland Europe and sold over 15 million copies worldwide. Stephen Thomas Erlewine of AllMusic called "Britney" "the record where she strives to deepen her persona, making it more adult while still recognizably Britney.