Thanks for response!
Unfortunately, I have to report that doing
sudo install-grub /dev/sda #/dev/sda is by device for hard-drive with lvm, may be I need some specific options here?
sudo upgrade-grub
do not cure boot for 4.4.0 kernel - same problem persists, still 4.3.0 kernel (in "Advanced GRUB menu") booting fine.
So I still assume the problem I's not in grub (config for both kernels 4.3.0 and 4.4.0 looks the same)
Can you please spot commands for GRUB that you are using (may be some specific options other them mine), do you use LVM?