the vectors ${{\boldsymbol \alpha}}, {{\boldsymbol \beta}}$ are called the fields intensities and, obviously, are constant “in space” as shown by the second group of equations known as the Poisson equations. For a double field we write $$\label{eq2_3}
  H^{(0)} = - {\varepsilon}_0 (\mathbf{r}_1  {\bdot} {{\boldsymbol \alpha}}+ \mathbf{r}_2  {\bdot} {{\boldsymbol \beta}})+{\rm const}$$ with constant vectors $\mathbf{r}_1,\mathbf{r}_2$. These vectors are pointing from $O$ to the centers of application of the two constant fields. The redundant parameter ${\varepsilon}_0 \geqslant 0$ allows the essential part of $H^{(0)}$ to vanish while $a^2+b^2 \ne 0$. In what follows, for the cases with $H^{(1)}$ independent of ${{\boldsymbol \alpha}},{{\boldsymbol \beta}}$ and non-constant $H^{(0)}$ we always put ${\varepsilon}_0 =1$.

If $\mathbf{r}_1 {\times} \mathbf{r}_2\ne 0$, then $\mathbf{r}_1, \mathbf{r}_2$ can be chosen to form an orthonormal pair and therefore can be taken as the first two vectors of the moving frame $$\label{eq2_4}
 \mathbf{r}_1= \mathbf{e}_1, \qquad \mathbf{r}_2=\mathbf{e}_2.$$ Note that for a large class of Hamilton functions including all considered here, without loss of generality one can suppose that ${{\boldsymbol \alpha}}$ and ${{\boldsymbol \beta}}$ are orthogonal [@KhRCD; @KhHMJ]. This fact will be used later in the change of variables which simplifies calculations and formulas.

We say that the system with the Hamilton function has 