d:1985] for hyperbolic systems of equations (\[eq:2.1\]) allows the construction of certain classes of $k$-wave solutions admitting $k$ arbitrary functions of one variable. The replacement of the matrix of derivatives $u_i^\alpha$ in the system of equations (\[eq:2.1\]) by the simple real element $L_i^\alpha$ allows us to construct more general classes of solutions by replacing the real elements with complex ones. A specific form of solution $u(x)$ of an elliptic system (\[eq:2.1\]) is postulated for which the tangent mapping $du(x)$ is a sum of a complex element and its complex conjugate

\[eq:2.4\] du\^(x)=(x)\^(u)\_i(u)dx\^i+|(x)|\^(u)|\_i(u)dx\^i,=1,…,q,

where $\gamma=(\gamma^1,\ldots, \gamma^q)\in{\mathbb{C}}^q$ and $\lambda=(\lambda_1,\ldots,\lambda_p)\in {\mathbb{C}}^p$ satisfy

\[eq:star\] \_i\_\^[i]{}(u)\^=0,|\_i\_\^[i]{}(u)|\^=0.

Here the quantity $\xi(x)\neq 0$ is treated as a complex function of the real variables $x$. In what follows we assume that the vectors $\gamma$ and $\bar{\gamma}$ are linearly independent. The proposed form of solution (\[eq:2.4\]) is more general than the one proposed in [@Jeffrey:1976; @Rozdestvenski:1983] for which the derivatives $u^\alpha_i$ are represented by a real simple element leading to a simple Riemann wave solution. To distinguish this situation from the one proposed in (\[eq:2.4\]), we call the real-valued solution associated with a complex element and its complex conjugate a simple mode solution in accordance with [@Perad:1985]. This means that all first-order derivatives of $u^\alpha$ with respect to $x^i$ are decomposable in the following way

\[eq:star2\] =(x)\^(u)\_i(u)+|(x)|\^(u)|\_i(u),

or alternatively $$\frac{{\partial}u^\alpha}{{\partial}x^i}=i{\left( \xi(x) \gamma^\alpha(u)\lambda_i(u)-\bar{\xi}(x)\bar{\gamma}(u)^\alpha\bar{\lambda}_i(u) \right)},$$where a set of functions $(\lambda,\gamma)$ and their conjugates $(\bar{\lambda},\bar{\gamma})$ on $U$ satisfy the wave relation (\[eq:star\]). Similarly, as in the case of $k$-waves, for hyperbolic sys