A 1981 model has points, located inside the flywheel. There are three main possible causes of your problem:
1. the kill wire is contacting ground somehow
2. the points are not working properly (need to be cleaned and reset)
3. The coil is unserviceable and needs to be replaced.
Post Edit: or 4. the ignition condenser is unserviceable and needs to be replaced.

You can simplify your ignition system, and reduce maintenance requirements, by fitting a Magnetron breakerless module in place of your breaker points ignition system. I have attached the official Briggs and Stratton instruction sheet for this rework, to this post, or you can download it from Briggs:

http://www.briggsandstratton.com/support/frequently-asked-questions/Replacing%20the%20ignition/~/media/9ADF0E828C7642A69BC9107A496A31A6.ashx

Joe Carroll has reported doing this successfully many times, using a less complicated procedure. You can probably find the thread where he describes how he does it.

There are two advantages of going this way: you don't have to remove the flywheel to check and adjust the points; and Magnetrons are likely to be a lot more common than breaker-point coils these days, being more recent.

If you decide to keep the engine original (retain the breaker points), I suggest you check the above four possibilities in whatever order you choose. If you get hold of a replacement breakerpoint-style coil, trying it may be easier than removing the flywheel to adjust the points. However pulling the flywheel is not all that hard if you make a simple puller and if you can get access to the American screws required, or if you have a 1/4" UNC tap you can use that, followed by ordinary 1/4" UNC screws.

Attachments
B&S magnetron_retro-fit.pdf (440.75 KB, 7 downloads)
Last edited by grumpy; 27/01/13 04:16 AM. Reason: Add post edit providing extra detail