Attempted to start , pushed start button , engine turned over maybe 3 seconds then stopped. pushed starter button again , Got no response from starter , thinking bad battery bought new one. still no response. Stop, run switch in RUN position. All other electrical systems operate fine. starting system completely inoperative.

Ok, make sure your battery is installed and connected correctly. The connections should be clean and tight. Now, look at the end of your starter. You'll see a five sided plate with a tower like thing in the middle on the end of your starter. Just above it is a single wire that plugs onto the starter. Unplug this wire and insert a test light into the wire. Turn your ignition on and press the start button. If the light lights up, your starter is bad. If it doesn, you've got another problem.

If the light does not light up, you'll have to find the starter relay. Harley moves these things around all the time so you never know where to look for one from year to year. I'd start under the seat and then move to the side panels if necessary. The relay is a typical one inch cube made of plastic. When you find it, turn on the igntion and press the start button while holding the relay. It should click. If if doesn't, it's probably bad and even if it does, it's probably bad. I'd replace the part as it's not all that expensive.

If you suspect the starter, go back and remove the three screws that hold the five sided plate onto the starter. Behind the plate you see a plunger with a spring on it. NOW, MAKE SURE THE TRANSMISSION IS OUT OF GEAR. Using a large screwdriver, push the plunger into the starter as far as it will go. The starter should engage and turn the engine. If not, you may have to replace the large copper contacts in the solenoid.

I hope this helps you find and fix your problem

Ride Safe
Steve