The aim of this paper is to classify the cubic polynomials

$$\begin{aligned} H(z,x,y)=\sum _{j+k\le 3}a_{jk}(z)x^jy^k \end{aligned}$$

over the field of algebraic functions such that the corresponding Hamiltonian system \(x'=H_y,\) \(y'=-H_x\) has at least one transcendental algebroid solution. Ignoring trivial subcases, the investigations essentially lead to several non-trivial Hamiltonians which are closely related to Painlevé’s equations \(\mathrm{P_{I}}\), \(\mathrm{P_{II}}\), \(\mathrm{P_{34}}\), and \(\mathrm{P_{IV}}\). Up to normalisation of the leading coefficients, common Hamiltonians are

$$\begin{aligned} \begin{array}{rl} \mathrm{H_I}:&{}-2y^3+\frac{1}{2}x^2-zy\\ \mathrm{H_{II/34}}:&{} x^2y-\frac{1}{2}y^2+\frac{1}{2}zy+\kappa x\\ \mathrm{H_{IV}}:&{}\begin{array}{l} x^2y+xy^2+2zxy+2\kappa x+2\lambda y\\ \frac{1}{3}(x^3+y^3)+zxy+\kappa x+\lambda y,\end{array} \end{array} \end{aligned}$$

but the zoo of non-equivalent Hamiltonians turns out to be much larger.

中文翻译：

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三次哈密顿量的Malmquist型定理

本文的目的是对三次多项式进行分类

$$ \ begin {aligned} H（z，x，y）= \ sum _ {j + k \ le 3} a_ {jk}（z）x ^ jy ^ k \ end {aligned} $$

在代数函数领域中，使得相应的哈密顿系统\（x'= H_y，\） \（y'=-H_x \）具有至少一个先验代数解。忽略琐碎的子情况，研究实质上导致了几个与Painlevé方程\（\ mathrm {P_ {I}} \），\（\ mathrm {P_ {II}} \），\（\ mathrm {P_ {34}} \）和\（\ mathrm {P_ {IV}} \）。直到前导系数的归一化，常见的哈密顿量为

$$ \ begin {aligned} \ begin {array} {rl} \ mathrm {H_I}：＆{}-2y ^ 3 + \ frac {1} {2} x ^ 2-zy \\ \ mathrm {H_ {II / 34}}：＆{} x ^ 2y- \ frac {1} {2} y ^ 2 + \ frac {1} {2} zy + \ kappa x \\ \ mathrm {H_ {IV}}：＆{} \ begin {array} {l} x ^ 2y + xy ^ 2 + 2zxy + 2 \ kappa x + 2 \ lambda y \\ \ frac {1} {3}（x ^ 3 + y ^ 3）+ zxy + \ kappa x + \ lambda y，\ end {array} \ end {array} \ end {aligned} $$

但是非等效哈密顿主义者的动物园却要大得多。