What is the best way for a trekking group of two to sleep in deep snow (meter or more). What equipment is to be brought?
Preferably, the technique should work on shallow snow and plain frozen ground as well.
Best is of course subjective, but overall, given your question and caveats, I would say a good solid winter tent.
To elaborate on the alternatives, if you have an assured amount of snow, then you can attempt some form of snow shelter. Building an actual igloo takes quite a bit of practice. Building a snow cave on the other hand is not that difficult, but does need quite a pile of snow of the right texture. Given that you want this method to work on plain frozen ground then any kind of snow shelter is likely out.
Other temporary shelters, like lean-tos also largely depend upon the surrounding area for building materials. If you are in more open country, you may well lack the requisite branches/trees, in order to make this work. Also depending on where you camp, you might get in trouble for stripping live branches for your shelter, or to sleep on, so know your area.
Overall, therefore, a good solid winter-rated tent is probably your best bet. You'll know you have it with you, and even if you want to try out a snow shelter or other variation for one night, you'll have the tent to fall back on.
Given your criteria, I would go with bivy sacks. Making shelters from snow (even just tarp covered windbreaks) is energy and time consuming, eating up your already scarce daylight and what little energy you have leftover from hiking/snowshoeing/skiing to camp.
A commentary on group size, though: it's almost always a bad idea to go winter mountaineering with less than three people, both for trailbreaking reasons and most winter (four season) tents are heavy and not made for less than three people.
No matter what you decide, you will need a snow shovel to carve out and help pack down a space for your shelter (or carve out your shelter). Snowshoes are also very handy for packing down snow to make a surface for a bivy sack or tent.