Group Classification of a Generalized Lane-Emden System

Table 1

Lie symmetries for 𝑛≠−1,1,3, for various functions 𝐻(𝑣) and 𝐺(𝑢).

---

𝐻(𝑣) arbitrary, 𝐺(𝑢)=𝑐, 𝑐 a constant

𝑋₁=𝜕𝜕𝑢,𝑋₂=𝑥^(1−𝑛)𝜕𝜕𝑢,  𝑋₃=(𝑐𝑥²𝜕+2𝑛𝑣+2𝑣)𝜕𝑢

---

𝐻(𝑣)=𝑑, 𝑑 a constant, 𝐺(𝑢) arbitrary

𝑋₁=𝜕𝜕𝑣,𝑋₂=𝑥^(1−𝑛)𝜕𝜕𝑣,𝑋₃=(𝑑𝑥²𝜕+2𝑛𝑢+2𝑢)𝜕𝑣

---

𝐻(𝑣)=𝑑, 𝑑 a constant, 𝐺(𝑢)=𝛼+𝛽𝑢, 𝛼 and 𝛽 constants (𝛽≠0)

𝑋₁=𝜕𝜕𝑣,𝑋₂=𝑥^(1−𝑛)𝜕𝜕𝑣,𝑋₃=(𝑑𝑥²𝜕+2𝑛𝑢+2𝑢)𝜕𝑣,

𝑋₄=𝛽𝑥^(3−𝑛)𝜕𝜕𝑣+(2𝑛−6)𝑥^(1−𝑛)𝜕𝜕𝑢,

𝑋₅𝜕=(2𝑛+2)𝜕𝑢−𝛽𝑥²𝜕𝜕𝑣,

𝑋₆=2𝑛²𝑥𝜕+8𝑛+6𝑛𝜕𝑥−4²𝑢+3𝑑𝑥²+4𝑛𝑢+3𝑢+𝑛𝑑𝑥²𝜕𝜕𝑢+(𝛽𝑑𝑥²−2𝛼𝑛−6𝛼)𝑥²𝜕𝜕𝑣,

𝑋₇=(16𝑛𝑥+4𝑛²𝜕𝑥+12𝑥)𝜕𝑥−(4𝑛+12)𝑑𝑥²𝜕𝜕𝑢+(8𝑛²𝑣+32𝑛𝑣+𝛽𝑑𝑥⁴𝜕+24𝑣)𝜕𝑣

---

𝐻(𝑣)=𝑑, 𝑑 a constant, 𝐺(𝑢)=𝛼+𝛽𝑢^−𝑝, 𝛼,𝑝, and 𝛽 constants (𝛽,𝑝≠0)

𝑋₁=𝜕𝜕𝑣,𝑋₂=𝑥^(1−𝑛)𝜕𝜕𝑣,

𝑋₃𝜕=(𝑛𝑥+𝑥)𝜕𝜕𝑥+(2𝑛𝑢+2𝑢)𝜕𝑢−(2𝑛𝑝𝑣−2𝑛𝑣+𝛼𝑝𝑥²𝜕+2𝑝𝑣−2𝑣)𝜕𝑣,

𝑋₄=(2𝑢+2𝑛𝑢+𝑑𝑥²)𝜕𝜕𝑣

---

𝐻(𝑣)=𝑑, 𝑑 a constant, 𝐺(𝑢)=𝛼+𝛽𝑒^−𝑘𝑢, 𝛼,𝑘, and 𝛽 constants (𝛽,𝑘≠0)

𝑋₁=𝜕𝜕𝑣,𝑋₂=𝑥^(1−𝑛)𝜕𝜕𝑣,𝑋₃=(𝑑𝑥²𝜕+2𝑛𝑢+2𝑢)𝜕𝑣,

𝑋₄𝜕=(2𝑛+2)𝜕𝑢−(𝛼𝑘𝑥²𝜕+2𝑘𝑛𝑣+2𝑘𝑣)𝜕𝑣

---

𝐻(𝑣)=𝑑, 𝑑 a constant, 𝐺(𝑢)=𝛼+𝛽ln𝑢, 𝛼 and 𝛽 constants (𝛽≠0)

𝑋₁=𝜕𝜕𝑣,𝑋₂=𝑥^(1−𝑛)𝜕𝜕𝑣,𝑋₃=(𝑑𝑥²𝜕+2𝑛𝑢+2𝑢)𝜕𝑣,

𝑋₄𝜕=(𝑥𝑛+𝑥)𝜕𝜕𝑥+(2𝑢+2𝑛𝑢)𝜕𝑢+(2𝑛𝑣−𝛽𝑥²𝜕+2𝑣)𝜕𝑣

---

𝐻(𝑣)=𝑎+𝑏𝑣, 𝑎 and 𝑏 constants (𝑏≠0), 𝐺(𝑢)=𝑐, 𝑐 constant

𝑋₁=𝜕𝜕𝑢,𝑋₂=𝑥^(1−𝑛)𝜕𝜕𝑢,𝑋₃=(𝑐𝑥²𝜕+2𝑛𝑣+2𝑣)𝜕𝑢,

𝑋₄=𝑏𝑥^(3−𝑛)𝜕𝜕𝑢+(2𝑛−6)𝑥^(1−𝑛)𝜕𝜕𝑣,

𝑋₅=𝑏𝑥²𝜕𝜕𝜕𝑢−(2𝑛+2)𝜕𝑣,

𝑋₆=(𝑏𝑐𝑥⁴−32𝑛𝑢−12𝑎𝑥²−24𝑢−8𝑛²𝑢−4𝑎𝑛𝑥²)𝜕𝜕𝑢−(4𝑐𝑛𝑥²+24𝑣+8𝑛²𝑣+32𝑛𝑣+12𝑥²)𝜕𝜕𝑣,

𝑋₇=(16𝑛𝑥+4𝑛²𝜕𝑥+12𝑥)𝜕𝑥+(8𝑛²𝑢+24𝑢+32𝑛𝑢+𝑏𝑐𝑥⁴)𝜕𝜕𝑢−(12𝑐𝑥²+4𝑐𝑛𝑥²)𝜕𝜕𝑣

---

𝐻(𝑣)=𝑎+𝑏𝑣, 𝑎 and 𝑏 constants (𝑏≠0), 𝐺(𝑢)=𝛼+𝛽𝑢, 𝛼 and 𝛽 constants (𝛽≠0)

𝑋₁𝜕=𝐹(𝑥)𝜕𝑣,𝑋₂𝜕=𝑊(𝑥)𝜕𝑢,𝑋₃𝜕=𝑢𝜕𝜕𝑢+𝑣𝜕𝑣,

𝑋₄𝜕=𝑏𝑣𝜕𝜕𝑢+𝛽𝑢𝜕𝑣

---

𝐻(𝑣)=𝑎+𝑏𝑣, 𝑎 and 𝑏 constants (𝑏≠0), 𝐺(𝑢)=𝛽𝑢^−𝑝, 𝛽 and 𝑝 constants (𝛽,𝑝≠0)

𝑋₁𝜕=(𝑏𝑥+𝑏𝑝𝑥)𝜕𝜕𝑥+4𝑏𝑢𝜕𝜕𝑢+(2𝑎+2𝑏𝑣−2𝑎𝑝−2𝑏𝑝𝑣)𝜕𝑣

---

𝐻(𝑣)=𝑎+𝑏𝑣, 𝑎 and 𝑏 constants (𝑏≠0), 𝐺(𝑢)=𝛽𝑒^−𝑘𝑢, 𝛽 and 𝑘 constants (𝛽,𝑘≠0)

𝑋₁𝜕=𝑏𝑘𝑥𝜕𝜕𝑥+4𝑏𝜕𝜕𝑢−(2𝑎𝑘+2𝑏𝑘𝑣)𝜕𝑣

---

𝐻(𝑣)=𝑎+𝑏𝑣^−𝑚, 𝑎,𝑚, and 𝑏 constants (𝑏,𝑚≠0), 𝐺(𝑢)=𝑐, 𝑐 a constant

𝑋₁=𝜕𝜕𝑢,𝑋₂=𝑥^(1−𝑛)𝜕𝜕𝑢,𝑋₃=(𝑐𝑥²𝜕+2𝑛𝑣+2𝑣)𝜕𝑢,

𝑋₄𝜕=(𝑛𝑥+𝑥)𝜕𝑥+(2𝑢−2𝑛𝑚𝑢+2𝑛𝑢−𝑎𝑚𝑥²𝜕−2𝑚𝑢)𝜕𝜕𝑢+(2𝑛𝑣+2𝑣)𝜕𝑣

---

𝐻(𝑣)=𝑏𝑣^−𝑚, 𝑚 and 𝑏 constants (𝑏,𝑚≠0), 𝐺(𝑢)=𝛼+𝛽𝑢, 𝛼 and 𝛽 constants (𝛽≠0)

𝑋₁𝜕=(𝛽𝑥+𝛽𝑚𝑥)𝜕𝜕𝑥+(2𝛼−2𝛼𝑚+2𝛽𝑢−2𝛽𝑚𝑢)𝜕𝜕𝑢+4𝛽𝑣𝜕𝑣

---

𝐻(𝑣)=𝑏𝑣^−𝑚, 𝑚 and 𝑏 constants (𝑏,𝑚≠0), 𝐺(𝑢)=𝛽𝑢^−𝑝, 𝛽 and 𝑝 constants (𝛽,𝑝≠0)

𝑋₁𝜕=(𝑝𝑚𝑥−𝑥)𝜕𝜕𝑥+(2𝑚𝑢−2𝑢)𝜕𝜕𝑢+(2𝑝𝑣−2𝑣)𝜕𝑣

---

𝐻(𝑣)=𝑏𝑣^−𝑚, 𝑚 and 𝑏 constants (𝑏,𝑚≠0), 𝐺(𝑢)=𝛽𝑒^−𝑘𝑢, 𝛽 and 𝑘 constants (𝛽,𝑘≠0)

𝑋₁𝜕=𝑘𝑚𝑥𝜕𝜕𝑥+(2𝑚−2)𝜕𝜕𝑢+2𝑘𝑣𝜕𝑣

---

𝐻(𝑣)=𝑎+𝑏𝑒^−𝑚𝑣, 𝑎,𝑚 and 𝑏 constants (𝑏,𝑚≠0), 𝐺(𝑢)=𝑐, 𝑐 constant

𝑋₁=𝜕𝜕𝑢,𝑋₂=𝑥^(1−𝑛)𝜕𝜕𝑢,𝑋₃=(𝑐𝑥²𝜕+2𝑛𝑣+2𝑣)𝜕𝑢,

𝑋₄=(𝑎𝑥²𝜕+2𝑢+2𝑛𝑢)𝑚𝜕𝜕𝑢−2(𝑛+1)𝜕𝑣

---

𝐻(𝑣)=𝑏𝑒^−𝑚𝑣, 𝑚 and 𝑏 constants (𝑏,𝑚≠0), 𝐺(𝑢)=𝛼+𝛽𝑢 (𝛽≠0)

𝑋₁𝜕=𝛽𝑚𝑥𝜕𝜕𝑥−(2𝛼𝑚+2𝛽𝑚𝑢)𝜕𝜕𝑢+4𝛽𝜕𝑣

---

𝐻(𝑣)=𝑏𝑒^−𝑚𝑣, 𝑚 and 𝑏 constants (𝑏,𝑚≠0), 𝐺(𝑢)=𝛽𝑢^−𝑝, 𝛽 and 𝑝 constants (𝛽,𝑝≠0)

𝑋₁𝜕=𝑚𝑝𝑥𝜕𝜕𝑥+2𝑚𝑢𝜕𝜕𝑢+(2𝑝−2)𝜕𝑣

---

𝐻(𝑣)=𝑏𝑒^−𝑚𝑣, 𝑚 and 𝑏 constants (𝑏,𝑚≠0), 𝐺(𝑢)=𝛽𝑒^−𝑘𝑢, 𝛽 and 𝑘 constants (𝛽,𝑘≠0)

𝑋₁𝜕=𝑚𝑘𝑥𝜕𝜕𝑥+2𝑚𝜕𝜕𝑢+2𝑘𝜕𝑣

---

𝐻(𝑣)=𝑎+𝑏ln𝑣, 𝑎,𝑚 and 𝑏 constants (𝑏≠0), 𝐺(𝑢)=𝑐, 𝑐 constant

𝑋₁=𝜕𝜕𝑢,𝑋₂=𝑥^(1−𝑛)𝜕𝜕𝑢,𝑋₃=(𝑐𝑥²𝜕+2𝑣+2𝑛𝑣)𝜕𝑢,

𝑋₄𝜕=(𝑛+1)𝑥𝜕𝑥+(2𝑛𝑢−𝑏𝑥²𝜕+2𝑢)𝜕𝜕𝑢+(2𝑛𝑣+2𝑣)𝜕𝑣

---