u = vf(Z) (A4)

 Z = x2 + px3 (A5)

 where p and v, are constants to be determined and f is an arbitrary function of Z

[14-15]. Substituting into Eqs. A1-A3, we obtain

 r, = r, df/dZ (A6)

 D,kvk =0 (A8)

 where

 ( = (Cyk2 + k3)k (A9)

 Dk =C2k2 + p(2k3 +3k2) + 23k3 (A10)

 For a non-trivial solution of vi, it follows from Eq. A8. that the determinant of Dik

must vanish. Therefore,

 Dk = 0 (All)

 where

 Dll = c1212 + p(c,213 +1312)+ pc1313 = 6 + C55

 D,2 = 1222 + p(c223 + 1322)+ C1323 =p(c46 +C25)

 D13 = C1232 + p(C1233 + C1332)+ p2C1333 = C46 +P235

 D21 = c212 + (2213 + 2312 )+ P2313 = p(c46 + 25)

 D22 = C2+22 + P(C2223 + c322)+ p c2323 = C22 + P2c44 (Al2)

 D23 = c2232 + C33 + c33)+ p2C2333 = P(C2 + 44)

 D1 = C3212 + p(3213 + 3312)+ p 3313 = 46 2 35

 D32 = c322 +p(32 + c + C3322) + p2c3323 = p(c23 + C44)