Solution
= Solution
A <fixed singularity of a complex differential equation> has a location determined by the coefficients, independently of the initial value. A <movable singularity of a complex differential equation> belongs to an individual continued solution and changes location when the initial value changes. For example, $w'=w^2$ has the movable <pole> $z=c$ in $w=-1/(z-c)$. A fixed exceptional location need not be singular for every solution.
The precise first-order form of the <Painlevé determinateness theorem> needed here is as follows. Let $w'=A(z,w)$ be rational in $w$, with coefficients <meromorphic> in $z$ on the chosen independent-variable surface. Algebraic coefficients can be made single valued on their own surface. Exclude the fixed coefficient singularities and exceptional fibres where a reduced numerator and denominator vanish together, working also in the dependent coordinate $v=1/w$. If a solution is continued along an arc $\gamma:[0,1)\to U$ with endpoint $z_*=\lim_{t\to1}\gamma(t)$ outside that exceptional set, then \b[$w(\gamma(t))$ has a definite finite or infinite limit]. Any singularity there is a <pole> or an <algebraic branch point>; there is no movable essential or logarithmic singularity. The standard rectifiable-arc formulation is included in this statement.
Here is a proof. First suppose a sequence along the arc has $(z_n,w_n)\to(z_*,w_*)$ at a point where the vector field is <holomorphic>. Choose a smaller closed bidisc about this point. Uniform bounds on the vector field and its dependent-variable <derivative> give, by the integral <contraction mapping> proof, one radius $r>0$ on which every solution starting sufficiently close to $(z_*,w_*)$ exists and remains bounded. For large $n$, the solution through $(z_n,w_n)$ is therefore defined on a fixed disc about $z_*$. The entire sufficiently late tail of $\gamma$ lies in that disc. Uniqueness identifies the original continued solution with this local solution throughout that tail, so the solution extends through $z_*$ and has a limit. The same argument applies to an ordinary point in the reciprocal coordinate $v=1/w$.
If the limit did not exist, consider the cluster set $K$ in the dependent-variable <Riemann sphere>:
$$
K=\bigcap_{s<1}\overline{\{w(\gamma(t)):s\le t<1\}}.
$$
Each closed tail is compact and connected, and these sets are nested; hence $K$ is nonempty, compact and connected. The preceding paragraph excludes every ordinary point from $K$. At $z_*$ the rational vector field has only finitely many nonordinary dependent values, in both coordinate charts: an entire fibre of singular points was excluded among the fixed exceptions. Thus $K$ is a connected subset of a finite set, so it is a singleton. Compactness then forces convergence to its unique point, contradicting the assumed lack of a limit. This proves determinateness.
It remains to determine the local type rather than merely the limit. At a finite limiting value $w_*$ which is a pole of the vector field, write the reduced equation as $w'=P/Q$. Since $z_*$ is not exceptional, $P(z_*,w_*)\ne0$, and the inverse equation
$$
\frac{dz}{dw}=\frac{Q(z,w)}{P(z,w)}
$$
is <holomorphic> near $(z_*,w_*)$. The <holomorphic dependence of ordinary differential equations on parameters> makes its local solution curves jointly <holomorphic> in $w$ and their initial $z$ values. In particular, the value at the transverse line $w=w_*$ is a <holomorphic> first integral labelling these curves. The original graph has constant label; taking its endpoint limit shows that it lies on the inverse solution through $(z_*,w_*)$. That inverse solution cannot be the vertical curve $z\equiv z_*$: this would require $Q(z_*,w)$ to vanish identically near $w_*$, an excluded fixed singular fibre. Consequently its <power series> has a first nonzero term,
$$
z(w)-z_*=c(w-w_*)^k+O((w-w_*)^{k+1}),\qquad c\ne0.
$$
Here $k\ge2$ at a vector-field pole. Factor the right side as $(w-w_*)^k$ times a nonvanishing <holomorphic function> and take its local $k$th root. This gives a new dependent coordinate with nonzero <derivative>, and the <holomorphic inverse function theorem> expresses $w$ as a convergent <power series> in $(z-z_*)^{1/k}$. Thus the singularity is an <algebraic branch point>. At an infinite limit, repeat the argument for $v=1/w$. An ordinary zero of $v$ gives a <pole> of $w$, while a finite ramification of $v$ gives an algebraic pole branch. This proves the asserted local types and the theorem.
For the given equation, the finite-coordinate rational field is
$$
A(z,w)=\frac{w}{(z+1)(w^2-z^2)}.
$$
The whole fibre $z=-1$ is singular in the displayed coefficient, and at $(z,w)=(0,0)$ the reduced numerator and denominator vanish together. Hence the finite fixed exceptional set is $\{-1,0\}$. Away from these two locations, the only possible finite dependent limits at a singularity are $w_* =\sigma z_*$, with $\sigma=\pm1$.
At such a point the inverse equation is
$$
z_w=\frac{(z+1)(w^2-z^2)}w.
$$
It is <holomorphic> when $z_*\ne0,-1$ and $w_* =\sigma z_*$. Its inverse solution satisfies $z_w(w_*)=0$, and differentiation gives
$$
z_{ww}(w_*)=(z_*+1)\left(1+\frac{z_*^2}{w_*^2}\right)=2(z_*+1).
$$
Therefore
$$
z-z_*=(z_*+1)(w-w_*)^2+O((w-w_*)^3),
$$
and
$$
\boxed{w(z)=\sigma z_*\;\pm\sqrt{\frac{z-z_*}{z_*+1}}+O(z-z_*).}
$$
These are genuine square-root <algebraic branch points>. The inverse solutions can be based at arbitrary $z_*\notin\{0,-1\}$ and then continued to nearby ordinary initial points. Their distinct branch locations require different initial data, so \b[these branch points are movable]. The signs specify the two inverse branches near each of the two values $w_* =\pm z_*$.
There are \b[no movable poles]. Indeed the reciprocal equation is
$$
v'=-\frac{v^3}{(z+1)(1-z^2v^2)}.
$$
For every finite $z\ne-1$ it is <holomorphic> at $v=0$, and $v\equiv0$ is its unique solution with zero initial value. A nonzero reciprocal solution therefore cannot reach zero there. This rules out both an ordinary pole and an infinite algebraic branch at a movable finite location. Together with the theorem, it exhausts the finite movable singularities.
The fixed point zero must not be mistaken for a generic movable collision. It is ordinary for solutions with $w(0)\ne0$. But solutions reaching $w=0$ can have a fixed square-root singularity. To construct them rather than just balance powers, set $z(w)=w^2h(w)$ in the inverse equation. The equation for $h$ is
$$
wh'=1-2h+w^2(h-h^2)-w^4h^3,\qquad h(0)=\tfrac12.
$$
Equivalently,
$$
h(w)=\int_0^1t\left[1+w^2t^2\{h(wt)-h(wt)^2\}-w^4t^4h(wt)^3\right]dt.
$$
On a small disc this is a <contraction mapping> of the closed ball $\|h-1/2\|_\infty\le1/4$: both its departure from $1/2$ and its Lipschitz constant are $O(r^2)$. Its unique <holomorphic> fixed point gives $z=w^2/2+O(w^4)$. Thus
$$
w(z)=\pm\sqrt{2z}\,(1+O(z))
$$
is a genuine branch at the fixed location zero. The zero solution itself extends through zero as a function, although the original displayed coefficient is undefined at $(0,0)$.
The point $-1$ can produce a fixed nonalgebraic singularity. For an explicit example take $z=-1+x$ real, $0<x<x_0<1$, and $w=iy$ with initial $y>2$. Put $q=y^2$ and $L=-\log x$. Then
$$
\frac{dq}{dL}=\frac{2q}{q+(x-1)^2}.
$$
As $L$ increases, $q$ increases, its derivative is bounded above by $2$ and bounded below by a positive constant, and therefore the solution exists for all large $L$ and $q\to\infty$. The derivative then tends to $2$, so $q/L\to2$. Consequently
$$
w(z)^2\sim2\log(z+1)\qquad(z\downarrow-1).
$$
This is unbounded but grows more slowly than every negative power of $z+1$. It cannot be a meromorphic pole even after a finite ramified substitution, demonstrating an actual fixed singularity with logarithmic growth. The determinateness theorem does not exclude such behavior at fixed exceptions.
Finally, on the independent-variable <Riemann sphere> use $t=1/z$. For finite $w$ the transformed equation is
$$
\frac{dw}{dt}=\frac{tw}{(1+t)(1-t^2w^2)},
$$
which is ordinary at $t=0$. At simultaneous $z=\infty$, $w=\infty$, however, the reciprocal field is
$$
\frac{dv}{dt}=-\frac{tv^3}{(1+t)(v^2-t^2)},
$$
with an indeterminate point at $(t,v)=(0,0)$. Thus infinity is an additional fixed exceptional location on the sphere, rather than an unavoidable singularity of solutions with finite limiting value. There are solutions escaping to infinity there: for real initial data $w(z_0)>z_0>0$, the difference $d=w-z$ stays positive, since its derivative becomes positive as $d\downarrow0$. Moreover $d'\le0$ when $d\ge1$, so $0<d\le\max(1,d(z_0))$. Such a solution continues for all positive $z$, with $w=z+O(1)\to\infty$.
\b[In the finite plane, the fixed exceptional locations are $0,-1$; all other solution singularities are movable square-root branches at $w=\pm z$, and there are no movable poles. On the sphere, infinity is also a fixed exception at the infinite dependent value.] Being fixed or movable describes the base location, not merely whether the coefficient contains the factors $w-z$ and $w+z$.