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The simple answer for most situations is: no. Head-to-head teams-of-four matches simply don't work for the NGS, which requires more than two tables in play. And most privately played matches don't use electronic scoring, which is another requirement. However, there will be situations where the NGS can be applied:
If you fulfill these requirements then you can score the matches as if you were scoring a multiple teams event, ensuring that the seating lineup is correct and the movement (which might be very basic) is correct. Then produce a USEBIO file and submit it to My EBU. There's one catch - you won't want to be charged for this event, as EBULeague will already charge you, so you need to submit it with the special rate '08', which tells us it's an event which is charged for separately.