The race trotted and found to be sound" means that a horse, specifically one that participates in races, was exercised by trotting (a two-beat gait) and showed no signs of lameness, stiffness, injury, or abnormal movement, indicating the horse is physically healthy and capable of its intended work. 

Breakdown of the phrase:
The race: Refers to a horse bred and trained for racing, often a Thoroughbred. 

Trotted: The horse was made to move at a trot, a four-beat gait where the horse moves its legs in diagonal pairs (e.g., left front and right hind) and has a moment of suspension. 

Found to be sound: This is a positive assessment by a qualified individual, often a veterinarian, indicating the horse is free from any physical problems that would prevent it from performing. 

Signs of a sound horse (and what would indicate unsoundness):
Movement: A sound horse moves with an even, natural rhythm and smooth strides. 

Gait: They will walk, trot, canter, and gallop normally without limping, head-bobbing, or stiffness. 

Purpose: The horse is capable of performing its intended job (e.g., racing, riding, or pasture life) without pain or risk of further injury. 

In short, the phrase describes a routine check where the horse's movement at a trot confirmed its physical health and ability to work. 

I got the concern of your question, this is generally done when a horse runs far below expectations. First the trainer asks the jockey if something went wrong in the race, if the jockey explains that despite his efforts the horse didn't respond as it should have. The trainer requests the Vet to have a check if anything is wrong with the horse. 

In such scenario the horse is trotted and reported by the vet. If it is sound, the report says " was trotted after the race and found to be sound". In short nothing is wrong with the horse only it didn't run as expected. 

In Indian navy 

The security incharge of the submarine is called as trot. 

Good morning,

Thanks all for your reply ,past  days I was understood after the race horse was some problem.

After the race, the trotter (a horse bred for trotting races) was examined and found to be sound, meaning it showed no signs of injury or lameness. This is a positive outcome for the horse's health and well-being.

Trotted means....To ride...To ride and find out, the horse is sound...

The concerned authorities/veterinary doctors will ask the jockey to trot(run) it to check for any abnormalities and if it runs good then it is declared sound, meaning it's in good health and shape.

It seems like you're describing a horse racing scenario. After the race, the trotter (a horse bred for trotting races) was examined and found to be sound, meaning it showed no signs of injury or lameness. This is a positive outcome for the horse's health and well-being.