On a class of functional equations

Differentiable considered class 𝐶4 function 𝑓1,2 : → 𝑅, ⊂ ⊂ 𝑅𝑛+1 × 𝑅𝑛+1: 𝑓1(𝑥, 𝑦) = ( (𝑥, 𝑦),𝑤), 𝑓2(𝑥, 𝑦) = { ( (𝑥, 𝑦), 𝑧), where, { - are functions of class 𝐶4, (𝑥, 𝑦) = (𝑥1,..., 𝑥𝑛, 𝑦1,..., 𝑦𝑛), = 𝑥𝑛+1 - 𝑦𝑛+1, = 𝑥𝑛+1 + 𝑦𝑛+1, and the following inequalities hold: 𝜕𝑥𝑖 ̸= 0, 𝜕𝑦𝑖 ̸= 0, 𝜕 ̸= 0, 𝜕𝑤 ̸= 0, 𝜕{ 𝜕 ̸= 0, 𝜕{ 𝜕𝑧 ̸= 0. The functions 𝑓1,2 are two-point invariants of the action of some Lie group in the space 𝑅𝑛+1. The criterion of local invariance of such an action for these functions leads to functional differential equations: ((𝑌 (𝑥))′𝑥𝑛+1 + (𝑌 (𝑦))′𝑦𝑛+1)'′𝑤 + (𝑌 (𝑥) - (𝑦))'′′ = 0, (1) ((𝑌 (𝑥))′𝑥𝑛+1 + (𝑌 (𝑦))′𝑦𝑛+1) ′𝑧 + (𝑌 (𝑥) + (𝑦)) ′′ = 0, (2) where '(,𝑤) = -𝜕 𝜕𝑤/𝜕 and (, 𝑧) = -𝜕{ /𝜕{ 𝜕. Theorem 1. In the neighborhood 𝑈(⟨𝑥, 𝑦⟩) the equation (1), where = = 𝑥𝑛+1 - 𝑦𝑛+1, 𝑌 ̸= const, '′𝑤 ̸= 0, has the following solutions: = 𝐶(𝑥1,..., 𝑥𝑛), ' = 𝑎( )𝑤 + 𝑏( ); = 𝑟𝑥𝑛+1 + 𝑐, ' = 𝑎( ) 1 + 𝑏( ); = 𝑟(𝑥𝑛+1)2 + 𝑐, ' = 𝑎( ) 1 + 𝑏( ); = cos(𝜔𝑥𝑛+1 + ) + 𝑐, ' = 𝑎( )ctg 2 + 𝑏( ); = 𝑟𝑒𝜔𝑥𝑛+1 + 𝑐, ' = 𝑎( ) 𝑒𝜔𝑤 - 1 + 𝑏( ); = ch(𝜔𝑥𝑛+1 + ) + 𝑐, ' = 𝑎( )cth 2 + 𝑏( ); = sh(𝜔𝑥𝑛+1 + ) + 𝑐, ' = 𝑎( )th 2 + 𝑏( ), where 𝑟, 𝑐, = const, 𝐶(𝑥1,..., 𝑥𝑛) ̸= const, 𝑎( ), 𝑏( ) - are functions of class 𝐶3, 𝑎( ) ̸= 0. Theorem 2. In the neighborhood 𝑈(⟨𝑥, 𝑦⟩) the equation (2), where = = 𝑥𝑛+1 + 𝑦𝑛+1, 𝑌 ̸= 0, ′𝑧 ̸= 0, has the following solutions: = 𝐶(𝑥1,..., 𝑥𝑛), (, 𝑧) = 𝑎( )𝑧 + 𝑏( ); = 𝑟𝑥𝑛+1 + 𝑐, = 𝑎( ) 1 + 2𝑐 + 𝑏( ); = cos(𝜔𝑥𝑛+1 + ), = 𝑎( )tg + 2 2 + 𝑏( ); = 𝑟𝑒𝜔𝑥𝑛+1, = 𝑎( )𝑒-𝜔𝑧 + 𝑏( ); = ch(𝜔𝑥𝑛+1 + ), = 𝑎( )th + 2 2 + 𝑏( ); = sh(𝜔𝑥𝑛+1 + ), = 𝑎( )cth + 2 2 + 𝑏( ), где 𝑟, 𝑐, = const, 𝐶(𝑥1,..., 𝑥𝑛) ̸= const, 𝑎( ), 𝑏( ) - are functions of class 𝐶3, 𝑎( ) ̸= 0.