Does 'asymptomatic hyponatremia' exist?

Yes: This is explained in my prior answer on this topic, but the summary is that if the hyponatremia occurs slowly, it , ay be entirely a symptomatic, and found simply as a laboratory finding. Still, the cause should be sought.

Definitely: Hyponatremia means blood level lower than normal (usually below 135 meq/l)); can be due lots of causes including meds (duiretics), liver, heart, kidney issues, starvation, anorexia nervosa, excess water intake, and others.Can be seen as lab artifact if high glucose, triglycerides mild cases may have no symptoms at all. More severe cases may have no symptoms if slow in development.

Happens a lot: Hyponatremia occurs when the serum sodium is below 135 meq. Therefore many patients who consume too much water or lose a lot of water through running or diuretics can have a sodium level below 135. In must of these instances, these patients are asyptomatic.