Welcome to the forum. Please add your location and your bike's model and model year to your profile so that you don't have to remember to tell us every time and we don't have to keep asking when you forget (see Forum Settings link in my signature).
And welcome to the world of antique vehicle ownership (they own us, not the other way around). Your bike is about 4 decades old and may or may not have had all of the maintenance necessary to keep it safe & reliable (some of which is not obvious if you aren't familiar with these bikes) so it is highly recommended to download the Factory Shop Manual for your model (available through the CX Wiki - link in my signature) and go through all of the service procedures, regardless of whether your bike has reached the specified mileage.
I also recommend looking on all rubber parts with suspicion because rubber does not age gracefully. I guess I don't need to tell you to check the date codes on your tires and replace them if they are over 5 years old or to replace the original rubber brake line with a modern stainless braided ones but don't forget things like the rad hoses and the boot between the engine and swingarm (they can crack on the bottom where you don't see it).
I'll leave it for others to comment on the inadvisability of converting a CX500 swingarm to single shock without adding serious reinforcement and lowering the front of the tank so that you no longer have access to all of the fuel in it but I will comment that it looks like the lighting no longer meets legal requirements (I don't see any turn signals or front marker lights and retaining all original lighting functions is a legal requirement in the US and Canada for vehicles used on public roads).
