  The Nash problem on arcs for normal surface singularities states that there
are as many arc families on a germ (S,O) of a singular surface as there are
essential divisors over (S,O). It is known that this problem can be reduced to
the study of quasi-rational singularities. In this paper we give a positive
answer to the Nash problem for a family of non-rational quasi-rational
hypersurfaces. The same method is applied to answer positively to this problem
in the case of E_6 and E_7 type singularities, and to provide new proof in the
case of D_n, n> =4, type singularities.
