  We propose a two-step Newton's method for refining an approximation of a
singular zero whose deflation process terminates after one step, also known as
a deflation-one singularity. Given an isolated singular zero of a square
analytic system, our algorithm exploits an invertible linear operator obtained
by combining the Jacobian and a projection of the Hessian in the direction of
the kernel of the Jacobian. We prove the quadratic convergence of the two-step
Newton method when it is applied to an approximation of a deflation-one
singular zero. Also, the algorithm requires a smaller size of matrices than the
existing methods, making it more efficient. We demonstrate examples and
experiments to show the efficiency of the method.
