Mazda 6 Service Manual: No.2 vehicle moves in n position

No.1 vehicle does not move in d, s, l ranges, or in r position

No.3 vehicle moves in p position, or parking gear does not disengage when p is disengaged

Other materials:

Mazda 6 Service Manual: Transmitter
NOTE (With theft-deterrent system) The hazard warning lights flash when the theft-deterrent system is armed or turned off. Refer to Theft-Deterrent System. (With the advanced keyless function) A beep sound can be heard for confirmation when the doors are locked/ unlocked using the k ...

Mazda 6 Service Manual: Intermittent concern troubleshooting
  Vibration Method 1. If malfunction occurs or becomes worse while driving on a rough road or when engine is vibrating, perform the steps below. Note There are several reasons vehicle or engine vibration could cause an electrical malfunction. Some of the things to check for are: ...