Icelandic name

Icelandic names differ from most current Western family name systems by being patronymic or occasionally matronymic: they indicate the father (or mother) of the child and not the historic family lineage. Iceland shares a common cultural heritage with the Scandinavian countries of Norway, Sweden and Denmark with the Faroe Islands. Icelanders, however, unlike other Nordics, have continued to use their traditional name system, which was formerly used by all Nordic countries except partly Finland. The Icelandic system is thus not based on family names (although some people do have family names and might use both systems). Generally, a person's second name indicates the first name of their father (patronymic) or in some cases mother (matronymic). According to Icelandic naming tradition, second names end in -son or -dóttir with few exceptions.

Question: does everyone in iceland have a surname ending in son
Answer: no