  In this paper we apply the ideas of New Q-Newton's method directly to a
system 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-Marquardt
algorithm, where we prove some new results on global convergence and avoidance
of saddle points.
  The second algorithm proposed here is a modification of New Q-Newton's method
Backtracking, 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 is
more suitable than New Q-Newton's method Backtracking itself, while currently
has better avoidance of saddle points guarantee than Levenberg-Marquardt
algorithms.
  Also, a general scheme for second order methods for solving systems of
equations is proposed. We will also discuss a way to avoid that the limit of
the constructed sequence is a solution of $H(x)^{\intercal}F(x)=0$ but not of
$F(x)=0$.
