Three. (20 points) In the Cartesian coordinate system $x O y$, a line segment $A B$ of fixed length $m$ has its endpoints $A$ and $B$ sliding on the $x$-axis and $y$-axis, respectively. Let point $M$ satisfy
$$
A M=\lambda A B(\lambda>0, \lambda \neq 1) .
$$

Do there exist two distinct fixed points $E$ and $F$ such that $|M E|, |M B|, |M F|$ form an arithmetic sequence? If they exist, find the coordinates of points $E$ and $F$; if not, explain the reason.
Three, let $A(a, 0)$ and $B(0, b)$. Then $a^{2}+b^{2}=m^{2}$. From $A M=\lambda A M$, we know that $M$ divides $A B$ in the ratio $\frac{\lambda}{1-\lambda}$.
Let $M(x, y)$. Then
$$
\left\{\begin{array}{l}
x=\frac{a}{1+\frac{\lambda}{1-\lambda}}=(1-\lambda) a, \\
y=\frac{\frac{\lambda}{1-\lambda} b}{1+\frac{\lambda}{1-\lambda}}=\lambda b .
\end{array}\right.
$$

Thus, $a=\frac{x}{1-\lambda}, b=\frac{y}{\lambda}$.
Therefore, $\frac{x^{2}}{(1-\lambda)^{2}}+\frac{y^{2}}{\lambda^{2}}=m^{2}$,
which simplifies to $\frac{x^{2}}{(1-\lambda)^{2} m^{2}}+\frac{y^{2}}{\lambda^{2} m^{2}}=1$.
This is the equation of the locus of point $M$.
Assume there exist points $E$ and $F$. Then
$$
\begin{array}{l}
|\boldsymbol{M E}|+|\boldsymbol{M F}|=2|\boldsymbol{M B}| \\
=2|((1-\lambda) a,(\lambda-1) b)| \\
=2 m|1-\lambda| .
\end{array}
$$

When $\lambda=\frac{1}{2}$, $1-\lambda=\lambda$. In this case, the locus of point $M$ is a circle, and there do not exist points $E$ and $F$ that satisfy the conditions.

When $0<\lambda<\frac{1}{2}$, $\lambda<1-\lambda$. In this case, the locus of point $M$ is an ellipse with foci on the $x$-axis, with the length of the major axis being $2(1-\lambda) m=2|M B|$. Points $E$ and $F$ are the foci of the ellipse, with coordinates
$$
\begin{array}{l}
E(-\sqrt{1-2 \lambda} m, 0) \text{ and } F(\sqrt{1-2 \lambda} m, 0) . \\
\text{When } \lambda>\frac{1}{2} \text{ and } \lambda \neq 1, \lambda>|1-\lambda|. \text{ In this case,}
\end{array}
$$

When $\lambda>\frac{1}{2}$ and $\lambda \neq 1$, $\lambda>|1-\lambda|$. In this case, the locus of point $M$ is an ellipse with foci on the $y$-axis, with the length of the major axis being $2 \lambda m \neq 2|M B|$, and there do not exist points $E$ and $F$ that satisfy the conditions.

In summary, when $\lambda \geqslant \frac{1}{2}$ and $\lambda \neq 1$, there do not exist points $E$ and $F$ that satisfy the conditions; when $0<\lambda<\frac{1}{2}$, there exist two fixed points $E$ and $F$ that satisfy the conditions, with coordinates
$$
E(-\sqrt{1-2 \lambda} m, 0), F(\sqrt{1-2 \lambda} m, 0) .
$$