First, welcome to OCC.........
The LGTX1050 came with a Kohler SV730-3044 engine. In the Kohler Service Manual, the White Wire going to a Connector, is the engine "kill wire". That goes to the Cub Cadet Wiring Harness and is Yellow. For the engine to run, no foot on the Brake Pedal, someone has to be in the Seat, and your Ignition Switch has to be in the Start Position. You may have any of these switches bad, and/or bad/shorted wiring.