Assuming the mower has the correct type igntion coil which is a tan color the engine the terminal that is on the back of the coil needs to be grounded to run and open to kill spark. If the coil was switched out with a coil for later model lawn boys which is grey the terminal needs to be open to run and grounded to kill spark. Clean terminals with fine sandpaper to remove corrosion and test with jumper wire from terminal to engine block an bypass switch. The switch may be bad internally which remains in the run postion at all times. If you dont care about looks just install a generic toggle swtich on the engine shroud connecting the coil terminal to block. Or a dealer should have a new proper switch to install for around $30. Get back to us on what you come up with

You're very lucky. They're many people who can't get their Lawnboy to run that would love to have your problem. A simple solution is to take a length of primary wire, srip both ends, remove the spark plug boot, place one end in the spark plug boot so that it contacts the electrode when the boot is replaced and tape the other end to the handle. When you want to kill the engine simply touch the stripped end near the handle to a suitable ground. Disclaimer: This solution of course negates rhe engine kill switch/brake feature of machines so equipped. However, many people in the 50's and 60's operated mowers like this and still have 10 fingers and 10 toes.