Saraswati River doesn't exist indeed, but why does the branch of Luni River include Saraswati river as its headwater near Pushkar in Rajasthan?

 “Saraswat” means water in solid form or a holy form rich with forever freshness.

So Saraswati river is a generic term, does not represent any specific river.

In prayag raj or Triveni Sangam, Saraswat represents the presence of Sukshma form of Glacier Solid Water.

If you trace the Ganges, up to Dev Prayag and beyond till origination of Alak Nanda- the clear blue color Glacier river, called Saraswati River short lived merging with Pancha Tara Streams at Mana village.

Learn from now-

Saraswat” means water in solid form from a glacier or a holy form rich with forever freshness.