Solid rubber bicycle tires
As the comments have comprehensively pointed out, solid tyres have some significant problems.

I recently fitted a solid tyre from Tannus on my bike with the same goals as yours. I only managed to last a few weeks before I couldn't deal with the problems and refitted my standard tyres.

Pros


No flat tyres!


Cons


Rough ride - the ride quality was awful. Jarring and rattling constantly.
Poor grip - The tyre didn't have anywhere near the grip of a good standard tyre. 
Somewhat loose attachment to the rim - there was a sensation of the tyre 'rolling' off the rim. It didn't ever roll right off and I don't think it would, but there was noticeable give where the tyre attached to the rim leading to this instability when cornering.
Flat spotting - I locked up the wheel to avoid a car in the first 10 minutes of fitting the tyre and it created a significant flat spot. I actually had to feather the flat spot with sandpaper because there was a very noticable 'bump' with every revolution of the wheel.


If you want to give them a try, this tyre from tannus would fit your bike. The item code WMI702H or WMI702S would both fit, but be a bit narrower than your existing tyre (32 rather than 38).