I'm... not certain this actually solves to one solution... I THINK it does? but... it's hard to tell. If you assume symmetry, it becomes tricky, but doable.

Reminder though, symmetry does NOT guarantee only one solution, and definitely doesn't guarantee that all possible solutions are symmetrical.

Spoiler: Fill and connect the corners. All symmetric.

Thanks.

Even with symmetry, a lot of possibilites ...

Many, many of solutions. I eventually couldn't be bothered any more. Rubbish puzzle