Solving of functional equations associated with the scalar product

The functional equations
$$\left[X\right]\frac{\partial \chi}{\partial \theta} + X_{n+1}(x^{n+1})\frac{\partial \chi}{\partial x^{n+1}} + X_{n+1}(y^{n+1})\frac{\partial \chi}{\partial y^{n+1}} = 0,$$
$$[X]\frac{\partial \sigma}{\partial \theta} + (X_{n+1}(x) - X_{n+1}(y))\frac{\partial \sigma}{\partial w} = 0,  [X]\frac{\partial \varkappa}{\partial \theta} + (X_{n+1}(x) + X_{n+1}(y))\frac{\partial \varkappa}{\partial z} = 0, $$ is solved in the paper. Here $[X] = \sum^{n}_{k=1}\bigl(\varepsilon_kx^kX_k(y) + \varepsilon_ky^kX_k(x))$, $x = (x^1,\ldots,x^n,x^{n+1})$, $\varepsilon_k=\pm1$, the equations are arising in the embedding problem of the space $\mathbb R^n$ with the inner product of the form $\theta = \varepsilon_1x^1y^1 + \cdots + \varepsilon_nx^ny^n$. In this problem, all kinds of functions $f = f(\theta,x^{n+ 1},y^{n+ 1}) $ are found that are two-point invariants of $n(n + 1)/2$-parametric group of transformations.