Table 2.

Null forms S, integrating factors R (or the modified integrating factor ) and first integrals I for the vector fields admitted by equation (3.1) using the Darboux polynomials given in equations (3.14)–(3.17).

vector fieldsnull forms (S)integrating factor (R)integrals
V1R11I11
R12I12
R13I13
R14I14
V2R21I21
R22I22I21
R23I23I21
R24I24I21
V3R31I31
R32I32I31
R33I33I31
R34I34I31
V4R41I41I21
R42I42I21
R43I43I21
R44I44I21
V5R51I51
R52I52
R53I53
R54I54I13
V6R61I61
R62I62
R63I63
R64I64I13
V8R81I81
R82I82
R83I83
R84I84
• *In the vector field V7, we use the Darboux polynomials for calculating the integrating factor R.