Table 1.

Null forms (S), integrating factors (R), characteristics (Q), vector fields (V), Darboux polynomials (F) and first integrals (I) admitted by equation (3.1).

SRaQ=XVFI
V1
V5
V6I1
V4
V2I2
V8
V3
V7
• aOne may note that when the pair (S,R) gives the null form and integrating factor for the first integral I, then the pair is also the null form and integrating factor for the first integral f(I) as shown in Chandrasekar et al.[5,26]. This fact is used to find the required R in this table.