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In this paper we apply the ideas of New Q-Newton's method directly to asystem of equations, utilising the specialties of the cost function$f=||F||^2$, where $F=(f_1,\ldots ,f_m)$.  The first algorithm proposed here is a modification of Levenberg-Marquardtalgorithm, where we prove some new results on global convergence and avoidanceof saddle points.  The second algorithm proposed here is a modification of New Q-Newton's methodBacktracking, where we use the operator $\nabla ^2f(x)+\delta ||F(x)||^{\tau}$instead of $\nabla ^2f(x)+\delta ||\nabla f(x)||^{\tau}$. This new version ismore suitable than New Q-Newton's method Backtracking itself, while currentlyhas better avoidance of saddle points guarantee than Levenberg-Marquardtalgorithms.  Also, a general scheme for second order methods for solving systems ofequations is proposed. We will also discuss a way to avoid that the limit ofthe constructed sequence is a solution of $H(x)^{\intercal}F(x)=0$ but not of$F(x)=0$.

A more direct and better variant of New Q-Newton's method Backtracking for m equations in m variables