{
  "abstract": "We develop a componentwise geometric and monodromy theory for the one-support generalized Lamé equation on an elliptic curve, with singularities at \\(0\\) and \\(\\pm p\\). Its log-free curve decomposes canonically into irreducible even and non-even components, the former being governed by elliptic Painlevé~VI. For the non-even component, we construct the hyperelliptic spectral curve, Baker--Akhiezer functions, and addition map, and prove that \\[ °σ_{n,p}^{(1)}=n(n+1). \\] After quotienting by the involution \\(T\\mapsto -T\\), we identify the non-even spectral curve with the classical Lamé spectral curve of weight \\(n\\), compatibly with the addition map and the rational function \\(κ\\). This identification is realized by \\[ \\widetilde B=T^2-n(n+1)\\wp(p), \\] and associates every non-even generalized Lamé equation with a unique classical Lamé equation on the same elliptic curve having equivalent period monodromy. Fixing \\(\\widetilde B\\) yields an isomonodromic deformation with \\(τ\\) fixed. Together with the Painlevé-VI deformation on the even component, it gives a componentwise interpretation of the collision \\(p\\to0\\), and yields a finite descent on the admissible completely reducible locus. The classical spectral, finite-gap, finite-monodromy, and curvature theories consequently transfer to the non-even component. Finally, within the symmetric family $\\left(n_0,n_1,n_2,n_3,\\frac12,\\frac12\\right),$ the one-support case forms an affine genus-zero hierarchy, whereas for $\\left(1,1,0,0,\\frac12,\\frac12\\right)$, the non-even normalization is generically elliptic and becomes rational on the discriminant locus, while the full compactified log-free curve retains arithmetic genus two. This first genus jump marks the boundary of the affine theory and motivates a genus-dependent componentwise geometry.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2608.22243",
  "author_total": 2,
  "authors": [
    {
      "name": "Ting-Jung Kuo"
    },
    {
      "name": "Xuanpu Liang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/fbtzjpv4k5mqn.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2608.22243",
  "fulltext_url": null,
  "id": "fbtzjpv4k5mqn",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2608.22243",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.AP"
    ],
    [
      "arxiv",
      "math.CA"
    ],
    [
      "arxiv",
      "math.DG"
    ],
    [
      "openalex",
      "3109|Nonlinear Waves and Solitons"
    ]
  ],
  "openalex": "W7204164887",
  "pdf_url": "https://arxiv.org/pdf/2608.22243",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-07",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/fbtzjpv4k5mqn.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:19:58Z",
      "native_id": "2608.22243",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2608.22243"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.DG",
      "subfield": "geometry-topology"
    }
  ],
  "title": "Componentwise Geometry and Monodromy of Generalized Lamé Equations",
  "title_text": "Componentwise Geometry and Monodromy of Generalized Lamé Equations",
  "type": "preprint",
  "updated_at": "2026-10-30T12:19:58Z",
  "url": "https://research.draconitic.com/paper/fbtzjpv4k5mqn",
  "venue": "arXiv",
  "versions": []
}