It was about this time with Alexander the 9th. that the spelling started to switchbetween Burnet and Burnett. Most historians use the double T's from that period on.
The family line continued as follows.

JOHN BURNETT inherited the tile from his Father on 2 Sep. 1574.
He was married to Elizabeth Lumsden.

ALEXANDER BURNETT (death was before May 1578)
He was married to Katherine Arbuthnot. ca 1560

ALEXANDER BURNETT (death was on 5 July 1619)
He was married to Katharine Gordon ca 1578

His son Thomas became 1st. Baronet when he succeeded his Father to the estate of Leys in 1619.

David Burnett
Canadian Ambassador
House of Burnett