## 2 Deflation Technique  
The approach proposed in the paper uses a concept known in mathematics as deflation [37; 38; 39; 40; 41]. The theoretical foundations of deflation methods were discussed in [37]. In [38], theoretical considerations were extended to the case of infinite-dimensional Banach spaces, allowing deflation techniques to be applied to systems of partial differential equations. Moreover, new classes of deflation operators were introduced. The purpose of paper [39] was to present a deflation algorithm for finding the roots of a system of nonlinear equations where the Jacobian matrix is singular at a root. A systematic and automatic way to reduce the degree of the multiplicity of roots was proposed. A deflation method to find the successive eigenvalues in the numerical simulation of the band structure of three-dimensional dispersive metallic photonic crystals with face-centered cubic lattice was presented in [40]. In [41], the continuation Newton method with the deflation technique and the quasi-genetic evolution for the global minimum searching of an optimization problem was established. One of the interesting problems when solving a nonlinear equation is the existence of a magnetic zero [37]. The magnetic zero is the zero (root) to which the algorithm converges almost independently of the initial guess selected. Such a magnetic zero can often mask zeros of real interest. A particular advantage of deflation techniques is the possibility of avoiding convergence to such a magnetic zero once it has been determined. The basis of the deflation method is the following technique [37]. After determining the root, a new equation (system of equations) is created so that it retains those roots of the original equation (system) that remain to be calculated, but the previously found root is no longer a solution of the equation. The deflation technique was studied initially in the one-dimensional case. Let \(f(x)\) be a scalar polynomial and \(x^{1},\,\cdots,\,x^{k}\) be roots of \(f(x)\) identified with some iterative algorithm, such as the Newton method using, e.g., different initial guesses. Different roots of \(f(x)\) may be found by considering the deflated function:  
\[g(x)=\frac{f(x)}{(x-x^{1})\cdot(x-x^{2})\cdot\,\cdots\,\cdot(x-x^{k})} \tag{1}\]  
and applying the same iterative algorithm to \(g(x)\) as to \(f(x)\). The paper [37] introduced the concept of the deflation matrix, developed deflation theory for systems of nonlinear equations, and extended it to the problem of determining multiple roots. Several specific deflation techniques were proposed, and their application to function minimization was shown. Let us consider a nonlinear equation:  
\[\mathbf{f}(\mathbf{x})=\mathbf{0}, \tag{2}\]  
where \(\mathbf{x}=[x_{1},\,\ldots,\,x_{n}]^{\mathsf{T}}\), \(\mathbf{f}(\mathbf{x})=[f_{1}(\mathbf{x}),\,\ldots,\,f_{n}(\mathbf{x})]^{ \mathsf{T}}\), and \(\mathsf{T}\) denotes the transpose. Suppose that (2) has several solutions and one of them, labeled \(\mathbf{x}^{1}\), has already been found. To determine other solutions, Equation (2) is deformed to retain the remaining solutions, but avoids \(\mathbf{x}^{1}\). In [37], the following deformation was proposed. Let \(\mathbf{M}(\mathbf{x},\,\mathbf{x}^{1})\) be an \(n\) by \(n\) matrix, called a deflation matrix, such that  
\[\lim_{i\to\infty}\inf||\mathbf{M}\Big{(}\mathbf{x}^{(i)},\,\mathbf{x}^{1} \Big{)}\mathbf{f}\Big{(}\mathbf{x}^{(i)}\Big{)}||>0 \tag{3}\]  
for any sequence \(\mathbf{x}^{(i)}\to\mathbf{x}^{1}\). Let us create the function:  
\[\mathbf{g}(\mathbf{x})=\mathbf{M}\Big{(}\mathbf{x},\,\mathbf{x}^{1}\Big{)} \mathbf{f}(\mathbf{x}). \tag{4}\]  
We solve Equation (4) using a method of determining a single solution (e.g., the Newton method). Such a choice of matrices that satisfy Condition (3) ensures that \(\mathbf{x}^{1}\) is not a solution of Equation (4) and that the sequence generated by the method will converge to another solution of the original Equation (2) or diverge.