the system idle Process is not a process, more a counter which is displayed in WinTasks used for measuring how much idle time the CPU is having at any particular time. This counter will display how much CPU Resources, as a percentage are 'idle' and available for use.
Cannot be killed
Non-system processes like system idle originate from software you installed on your system. Since most applications store data in your system's registry, it is likely that over time your registry suffers fragmentation and accumulates invalid entries which can affect your PC's performance. It is recommended that you check your registry to identify slowdown issues