Lie Group Classification of a Generalized Lane-Emden Type System in Two Dimensions

Table 1

Classification results: 𝑘≠0  (𝑝=0) and 𝑘=0  (𝑝≠0).

𝑓	𝑔	Condition on const.	Extra operator(s)
𝑓0𝑒^−𝜆𝑣	𝑔0𝑢^−𝛾	𝜆≠0	𝑋4=𝜆𝛾𝑥𝜕𝑥+𝜆𝛾𝑦𝜕𝑦+2𝜆𝑢𝜕𝑢+2(𝛾−1)𝜕𝑣
	𝑔0	𝜆≠0	𝑋4=𝜆𝑢𝜕𝑢−𝜕𝑣,  𝑋𝑐=𝑐(𝑥,𝑦)𝜕𝑢
𝑓0𝑣^−1	𝛿ln𝑢	𝛿=±1	𝑋4=𝑥𝜕𝑥+𝑦𝜕𝑦+2𝑣𝜕𝑣
	𝑔0𝑢^−𝛾	𝜇=±1	𝑋4=(1+𝛾)𝑥𝜕𝑥+(1+𝛾)𝑦𝜕𝑦+4𝑢𝜕𝑢+2(1−𝛾)𝑣𝜕𝑣
𝜇𝑣	𝑔0𝑒^−𝜈𝑢	𝜈≠0	𝑋4=𝜈𝑥𝜕𝑥+𝜈𝑦𝜕𝑦+4𝑢𝜕𝑢−2𝜈𝑣𝜕𝑣
	𝜌𝑢	𝜌=±1	𝑋4=𝑢𝜕𝑢+𝑣𝜕𝑣,  𝑋𝑑=𝑑(𝑥,𝑦)𝜕𝑣
𝛽ln𝑣	𝑔0	𝛽=±1	𝑋4=𝑥𝜕𝑥+𝑦𝜕𝑦+2𝑢𝜕𝑢+2𝑣𝜕𝑣,  𝑋𝑐=𝑐(𝑥,𝑦)𝜕𝑢
𝑓(𝑣)	𝑔0𝑢^−1		𝑋4=𝑥𝜕𝑥+𝑦𝜕𝑦+2𝑢𝜕𝑢
	𝑔0		𝑋𝑐=𝑐(𝑥,𝑦)𝜕𝑢