Strategies for Newton's Method when the Jacobian at the solution is singular

I'm trying to solve the following system of equations for the variables $P,x_1$ and $x_2$ (all else are constants):

$$\frac{A(1-P)}{2}-k_1x_1=0 \\ \frac{AP}{2}-k_2x_2=0 \\ \frac{(1-P)(r_1+x_1)^4}{L_1}-\frac{P(r_1+x_2)^4}{L_2}=0$$

I can see that I can turn this system of equations into a single equation of a single variable $(P)$ by solving equations 1 and 2 for $x_1$ and $x_2$ respectively and substituting them into equation 3. In doing so, I am able to use matlab's fzero command to find the solution. Using the parameters $k_1=k_2=1$, $r_1=r_2=0.2$, and $A=2$, I found the true solution to be $P=x_1=x_2=0.5$.

However, when I use newton's method applied to the original 3 variate - 3 equation system, the iterations never converge to the solution, no matter how close I begin to the true solution $x^*=(P^*,x_1^*,x_2^*)=(0.5,0.5,0.5)$.

At first, I suspected my a bug in my implementation of newton's method. After checking several times, I found no bug. Then I tried using an initial guess $x_0=x^*$, and lo & behold: the Jacobian is singular. I know that a singular jacobian can reduce the order of convergence, but I don't think it necessarily prevents convergence to the true solution.

So, my question is, Given that the jacobian of the system at the true solution is singular:

1. What other conditions are necessary to prove that newton's method will not converge to the root?

2. Would a globalization strategy (e.g. line-search) guarantee convergence despite the singular jacobian?