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# SN~ (~6) lff 2It 3k_~ , 5 ",,x_J {b)

SN~ (~6) lff 2It 3k_~ , 5 ",,x_J {b)

## We define Au~(p, q) = p=

We define Au~(p, q) = p= qv - P~ q.. 188 b = qoPu2 - qt, P,, -P,,Pt, e = p~xAaa,(p,q)- 2pxpaAoax(p,q) + p~aAax,(p,q ) d = Zxo.(p,q)a..(p,r) - ao.(p, Oa.~(p,q) e = Aa,,(p, q) AaX(p , r) - Aa,,(p , r) Aax(p, q) g = 2 6'/3 IP,, I - ~ I po, I-½ I qo I ~ (p.o .-4,~,.x h=p~ Iqol -~ po, I qo case III (n = 3) (Pa=Pu" - Pan) el = signpa~(pa2px2 - p2ax ) case IV (n = 4) e2 = sign(h), u = sign(qob) (n = 5, > 7) ,2 = sign(j) case V (n = 4) ~ = sign[qo(pa~qo - p.qo)] case VI (Vn) el = sign(\$Aax(p,q)) case VII (n = 3) ~1 = sign(pa c) case IX (Vn) / el = sign(pa Aax(p,q)), e~ = sign(pa Aau(p,q)) t ~a = sign(e~ e) , ~4 = sign(~2 d) 7~(f) contains Table 2. Coefficients of the normal forms. - - 2 puqa) n--2 qo I (+m2, m,E1) IT (n=3)(+m+m3, m,El) (n>4)(< u"-l,v> + < u,v,A> m+m s, m, E1 ) III (n=3)(+m3, m,E1) (n > 4)(< un-l,v > + < u,v > m+ m3, m, E1) /V (n=4)(m+m 3,+m ~,m) (n--5,6)(< v,u 4,5 > Jr mq-ra 3, m, El) (n > 7) (< v,u "-1 > m+ < A >2 + < A > m 2 +m4,< v,A,u "-1 > +m 2, m) V (n=3)(+m+m4, m,E1) (n > 4) (< un-l,v > + < u,v,A >2 + < u,v,A > m 2 + rn4,< u,v,A > +m 2, E1 ) VI ( < u, v > + m2, < u, v > + m 2, m ) VII ( < u, v > + ma, < u, v > + m 3, m) V//Y (+m+mn,+m ~,rn) /X (rn 2,m 2,m 2) Table 3. Algebraic data.

qo r0 =0 =O #o ~o Pa. ~o 1=° top-cod> 2 ~Pa 1--o Pa= --0 Pa~P~ 1--o top-cod_~ 2 Pa ----0 q~ p ~ p~ ( 2p~po~ - q~ ) top-cod_~ 2 ~o #o e d A~u(p, q)Aa~(p, q) 1--o top-cod> 2 189 ~o , Px 2 po=p~2 - p~ 1=° top-cod> 2 ~o • Aax(p, q) 1--o g I =° top-cod> 2 ~o Figure 5: Flowchart for n -- 3. ~o ~o ~o ~o ~o .I • II ,III .V . VI -, VII ,.,IX VIII

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