Preamble

This work, originating from 1967, examines the differential equation of the form:
$$zww'' = zw'^2 - ww' + aw^3$$
and its related systems [2]. We consider the parameters $\alpha, \beta, \gamma, \delta$ and the system (1). By substituting the expressions for $u$ and $v$:
$$\begin{aligned}
u &= a_0(z) + a_1(z)w + a_2(z)w^2 + a_3(z)w^3 \
v &= b_0(z) + b_1(z)w + b_2(z)v + b_3(z)wv + b_4(z)wv^2
\end{aligned}$$
into the governing equations, we derive the relationships between the coefficients $a_j(z)$ and $b_j(z)$. Specifically, we find that $-a_0 a_1 = -(\beta - a_0)$, $b_3 + a_2 = 0$, $a_1 + b a_3 = \gamma$, and $b_4 + b_3 = 0$. Given $a_3 = \delta$, $b_1 = 0$, and $b = 0$ in system (1), where $b \neq 0$ and $a_3 \neq 0$, the condition $a^2 - \delta a_3 + \beta a_3 + b_2 a_3 = \gamma$ must be satisfied.

Following the methodology of N. A. Lukashevich, we consider the transformation $u = zw$, which leads to:
$$\begin{aligned}
z \frac{dw}{dz} &= a_0 z + hw + w^2 u \
u &= a z + \gamma z^2 w - ha - wu^2
\end{aligned}$$
For the case where $b \neq 0$, system (1) can be rewritten as $z \frac{dw}{dz} = aw + zw \pm zw$. The relationship $(a-1)ww' + zw^2$ and $a = -1, w = \phi(z, a, \beta, \gamma, \delta)$ for system (1). In (1), let $z' = e^{2t}$. If $8w^2 \neq 0$, then from (7) and (1) we have:
a) $w = \phi^{-1}(z, -\beta, -\alpha, -\delta, -\gamma)$,
b) $w = \phi^{-1}(-z, \beta, \alpha, -\delta, -\gamma)$,
c) $w = -\phi^{-1}(z, \beta, \alpha, -\delta, -\gamma)$,
d) $w = \phi^{-1}(z, -\beta, -\alpha, -\delta, -\gamma)$, and $8w' = e^t = 1$.
At $z = 0$, for cases a) through d), we observe that $z=0$ is a singular point. We note that at $z=0$, the condition $\alpha = \gamma = 0$ holds. Thus, at $z=0$, we have the expansion $a_0 \phi + a_2 z^2$ for (1) where $z, \alpha, \gamma$ satisfy $(1 - 3a_{-1} - 4\gamma a_{-1}) a_0 = 0$ and $(4 - 3\alpha a_{-1} - 4\gamma a_{-1}) A = 3(9 - 3\alpha a_{-1} - 4\gamma a_{-1}) a_{n-1} + \alpha(a_n + b a_{n-1}) + \gamma(4a_{-1} a_0 + 12 a_n) + \beta a_0 [(n+1)^2 - 3\alpha a_{-1} - 4\gamma a_{-1}] a_{n-1} a_n = P_n(a_{-1}, a_0, a_n)$ for $n=0, 1, 2, \dots$.
The coefficients $a_n$ are given by:
$$[(n+1)^2 - \alpha^2] a_{n-1} P_n(a_{-1}, a_0, \dots, a_{n-1}) = a_n(\alpha, \beta, \delta) \quad (10)$$
From (10), at $z=0$ for system (1), if $k^2 \gamma - \alpha^2 \neq 0$, then $a_{n-1} = 0$, and for $j=0, 1, 2, \dots$, we have $k^2 \gamma - \alpha^2 = 0$. In (8), the coefficients $a_j$ are determined. Setting $wu=1$ in (6), we obtain $z - \alpha u - z u w_x$ and $z u w^2$. Let $a = r = 2\gamma$ and $x, \alpha, v$ satisfy (11). Then $w_1 = J_x y + v = b + v$.
From $2\sqrt{\gamma} - \alpha z - z(b + v) \alpha^2$, we get $z = (a-1)v + zu[b + (b + \alpha)]$ (12).
As $z \to 0$, we have $(u, v) \to 0$ if $a_{n-1} = 0$, meaning $\alpha$ is not an integer. If $\alpha$ is an integer, then $(u, v) \to 0$ as $z \to 0$. Following N. A. Lukashevich for $z=0$ [3], from (8) we have $a = 2\sqrt{\gamma}$. If $\beta \neq 0$ and $\beta = 0$, then $(w, u) \to 0$ as $z \to 0$. If $\beta = \delta = 0$, then $w \to 0$ as $z \to 0$. If $\alpha = \gamma = 0$, then for system (1) at $z=0$, we have the series (13).
In (1) with $a_0 = 0$ and $\beta \delta \neq 0$, the coefficients $a_j$ ($j=1, 2, \dots$) satisfy $[(n+1)^2 \delta + \beta^2] a_n = \sigma_n(\alpha, \beta, \gamma, \delta)$. If $n+1 + \beta^2 = 0$, then for $j=1, 2, \dots$ where $25 + \beta^2 \neq 0$, the series (13) depends on $\beta$ and $\delta$. If $a_0 = 0$, then (13) yields $w=0$. In (13), $a_0$ is determined by $k^2 \delta + \beta^2 = 0$.
We have three cases: 1) $w(0) = 0$; 2) $w(0) = w'(0) = 0$; 3) $a_{n-1} = 0$. If $\beta \neq 0$ and $\delta = 0$, then $a_j$ are determined with $a_0 \neq 0$. For $j=1, 2, \dots$ in (13), if $\delta = 0$, then for system (1), $a_{n-1} = 0$. In case 3), at $z=0$, if $k^2 \delta - \beta^2 = 0$ and $a_{2s} = 0$ for $\alpha, \gamma, \delta$ in (1), let $z = x^s$ ($s=0, 1, \dots$). Then $w(z) = hz^3 + 2\alpha/z$ (15).
The equation $ww'' = \tau w'^2 - ww' + 9\tau^2(\alpha w^s + \gamma u - \sqrt{\gamma} x^s w_1 u^2) = 3[(\alpha-1)w_1 + \delta \tau^3 u + \tau^3 u w_1]$ holds. At $z=0$, following N. A. Lukashevich, (15) and (16) imply that if $a_0 \neq 0$, then $a_j$ depends on $\alpha, \beta, \gamma, \delta$. If $a_4 = a_1 = 0$, then $a_j = 0$ for $j > 4$. If $a_3, \delta$ are parameters of (1), then at $z=0$, if $a_0 = 0$, $a_j$ depends on $\alpha, \beta, \gamma, \delta$. If $z=0$, $x = a_2 = 0$, then $a_1 \neq 0$. In (16), if $a_2 = a_3 = 0$, then $a_2 \neq 0$ and $a_j, a_3 v = 0$, meaning $a_j = 0$. If $\beta = \delta = 0$ and $a_1 \neq 0$, then for (1), at $z=0$ according to [1, 4], $w(z) = 1/z$ and $v(z)$ satisfies (2).
We have $u = z u'' - u u' - \gamma z v^2 - \alpha u v$. At $z=0$, we obtain $z w'' = z v' - v v' + \delta z u^2 + \beta u v$. For $z_0 \neq 0$, $w(z) = \pm \frac{1}{\sqrt{\gamma}(z - z_0)}$. Following N. A. Lukashevich, with $h(z - z_0)$ and (7), system (1) yields $\Gamma = -1/w + z w^2 - \gamma w^2$. From (7) and (1), if $w_1$ is known, then $w_2$ satisfies $D(w_1, w'1, w''_1) = (\delta + w_2) z^2 w''_1 + z w'_1 w'_1 - z^2 w_1 w''_1 + 8 z w'_1 + \sqrt{\gamma} z^2 w'_1 + \beta - \alpha (\dots)$.
From (23), for $w, u, w_1, w_2$ as in [5], the dimension is $d=4$. If $w$ is known, then $D(w_2, w'_2, w''_2) = 0$ in (23). From (22), we find $d=4$. If $D(w_2, w'_2, w''_2) = 0$, then for (1), system (1) with (23) yields $\delta z w^2$ and $z w - 2 w_1 - \beta \neq 0$.
In the case $\sqrt{\gamma} w_1 + \delta z + z w_1 z w' + \dots$, we have $z w' + 1 - \dots + 2 w_2 - \beta$. If $w_1(z)$ satisfies (1) and $D(w_2, w'_2, w''_2) = 0$, then from (23), $w_1(z_0) = w$, $w(z_0) = w'(z_0) = w'}$. For (1), $w(z) - \rho(z_0) = w_{100$. From (24) and (25), if (28) and (29) hold, then $-\sqrt{\gamma} w_1 + (1 - \dots) z_0(w_2 w_0)$.
If $w_2(z)$ satisfies $D(w_2, w'_0, w''_2) = 0$ with $w_2(z_0)$ and $w'(z_0)$, then $w(z)$ satisfies (26) and (27). If (1) holds, then from (32), $w = w'0 + \sqrt{\gamma} w^2 + (\dots + 1) w_0 v = z_0 w'{20} + (\dots + 2) a$. At $z_0$, $w(z) = \dots$ and $w_1(z)$ at $z = z_0$. Following N. A. Lukashevich, (21) in (22) shows that $w_2(z)$ at $z_0$ has a pole.} - \beta$ for (1). In (7), at $z = z_0$, $w_1(z) = \frac{-\sqrt{\gamma}}{(z - z_0)^2} + \frac{1}{z - z_0
Thus, if $\gamma \neq 0$, $\alpha - 2 w_1 - \beta \neq 0$, and $(\dots + 2) w_2 - \delta \neq 0$, then for (1) at $z=0$, we have the expansion. We find $z w'_1 - (\dots - 2) w_1 - \beta = 0$. From (7), we see that $z w'_1$ satisfies the condition where $\sqrt{\gamma} + \delta = 0$.
Given $\rho, \gamma, \delta$ and $\rho \pm (w''_2) \sqrt{\delta} = \alpha$, then $z w' (\dots + 2) w_2 - \beta = 0$ and $w_2 \sqrt{\delta} \pm (w+2) \sqrt{\delta} = 0$. From (36) and (37), we have $\rho + \dots - \delta = 0$ and $\gamma = \alpha^2 \neq 0$. From (38) and (7) and (22), according to [2], $z = \alpha^ z w^ - (w + b^* z)$. In the case where $\gamma = 0$ and $\alpha \neq 0$, at $z = z_0 \neq 0$, system (1) is consistent with [5], so $d=4$. Substituting (19) into (1), we get:
$z u^2 v v'' - z v^2 u u' + z v^2 u'' - z u^2 v'^2 + u^2 v v' - v^2 u u' - \alpha v^3 u - \beta v u^3 - \gamma z v^4 - \delta z u^4 = 0$.
Let $v(z) = \sqrt{\gamma} u(z) \neq 0$. In (40) and (39), for $z$: a) $n=m$; b) $\gamma = \alpha = 0$; c) $n < m$ and $\beta = \delta = 0$.
In cases b) and c), we have $\alpha = \pm k \sqrt{\gamma}$. If $k > 0$ is not an integer and $\delta = 0$, $\alpha \gamma \neq 0$, then (1) has the solution $w(z) = C z^{1+k}$. In case a), $n=m-0$ for (1), so $w=0$, $\delta=0$, and $w = \pm \sqrt{-\beta} \neq 0$ with $\beta^2 + \alpha^2 = 0$.
For $z$ and $w$ in terms of $\beta, \gamma, \delta$, if $\phi < 0$ and $\beta^2 + \delta \alpha^2 = 0$, then $|u \gamma(z_0)| < n = m = 1$, i.e., $v = z + a$ and $u = bz + c$ at $z=0$. For $a, b, c$, if $b^4 = 0$, $b \neq \beta$, $b^3 + 4\gamma a + 4b b^3 c = 0$, $b \neq \alpha$, $c + 3\beta b^2 c + \beta a b^3 + 6\alpha^2 \gamma + 6b \sqrt{\delta} = 0$, $\alpha^2 - 3c - 3\beta b c^2 - 3\beta a b^2 c - 4\gamma \alpha^3 - 4\delta b c^3 = 0$, and $\alpha^3 b + \alpha^2 c + \beta c^3 + 3\beta a b c^2 + \gamma \alpha^4 + \delta c^4 = 0$, then $(c - ab - \alpha a^2 - \beta c^2) \alpha c = 0$.
Following N. A. Lukashevich, if $c \neq 0$, then: 1) $\alpha = (4b \delta - \beta) b^2$; 2) $\alpha = 0$ from (41); 3) $\alpha \gamma = -6b^4$ and $c=0$.
We have $S = (2ab + A)\delta$, where $S = \beta, \gamma, \epsilon$ are determined by (41). If $b^2(3c - ab) \epsilon$ satisfies (1) and $w(z)$ is such that $a = 0$ and $c \neq 0$, then for $\beta \neq 0$, we obtain $9\gamma - a^2 = 0$, $b' + \beta^2 = 0$, and $b^2 = -c$.

In the instance where $a \neq 0$ and $c = 0$, system (1) yields $\gamma - a^2 = 0$. Furthermore, if $9b + a = 20$ and $\beta^2 = 0$, we find $b^2 = -\dots$, $a^2 = -\dots$, and $\beta = 0$. Let us introduce the substitutions $u(z) = \xi(z) \exp g(z)$ and $v(z) = \eta(z) \exp g(z)$, where $\xi(z)$ and $\eta(z)$ satisfy the conditions derived from (43) and (20), and $g(z)$ is a function such that $\exp 2g(z)$ relates to the system dynamics. The resulting differential equations for $\eta$ are:
$$z \eta \eta'' = z \eta'^2 - \eta \eta' - [g'(z) + zg''(z)] \eta^2 + \delta z \xi^2 + \beta g \eta$$
By substituting (45) and (44) into the above, we obtain $g'(z) + zg''(z) = 2\lambda z + \mu$, which implies $g(z) = \lambda z + \mu$, where $\lambda$ and $\mu$ are constants.

The parameters $\alpha, \beta, \gamma, \delta$ are determined by the consistency of system (1) with equations (45) and (46). For the coefficients $a_n$ and $b_n$, we derive the following recurrence relations:
$$\begin{aligned}
z^{2n} + (2K a_n^2 + \gamma b_1^2) &= 0 \
b_1^2 + S a_1^2 &= 0 \
n_{-1} + 2\gamma &= -(\mu a_n) a_{n-1} - 4\lambda b = (-\beta a \mu b_n) b_n
\end{aligned}$$
Assuming $b_n \neq 0$, these satisfy (47) and (1). We then find $\lambda = \pm \sqrt{-\gamma}$ and $4\lambda^2 + \gamma^8 = 0$. For $k=2, 3, \dots$, the general form is:
$$z^{2n+1-k} 4k a_n a_{n-k} + 2\gamma b_n b_{n-k} = -P_k(n, a_{n-1}, a_{n-k+1}, b_n, b_{n-k+1})$$
where $P_k$ is a polynomial function of the indicated variables. The coefficients $a_j$ and $b_j$ are determined for all $j=0, 1, 2, \dots$ by (51). If $n \neq 1$ and $b \neq 0$, the parameters of (1) are constrained such that $\gamma = 0$ and $w(z)$ can be expressed in terms of $z$.

References

1. Lukashevich, N. A., Differentsial'nye Uravneniya, Vol. 7, No. 2, 1958.
2. Erugin, N. P., Vychislitelnaya Matematika, 1960, p. 21.
3. Golubev, V. V., Lectures on the Analytical Theory of Differential Equations, 1939.
4. Gambier, B., Acta Mathematica, Vol. 28, No. 2, 1912.