The Zhdanov–Trubnikov equation describing wrinkled premixed flames is studied, using pole decompositions as starting points. Its one-parameter (−1⩽c⩽+1) nonlinearity generalises the Michelson–Sivashinsky equation (c=0) to a stronger Darrieus–Landau instability. The shapes of steady flame crests (or periodic cells) are deduced from Laguerre (or Jacobi) polynomials when c≈−1, which numerical resolutions confirm. Large wrinkles are analysed via a pole density: adapting results of Dunkl relates their shapes to the generating function of Meixner–Pollaczek polynomials, which numerical results confirm for −1<c⩽0 (reduced stabilisation). Although locally ill-behaved if c>0 (over-stabilisation) such analytical solutions can yield accurate flame shapes for 0⩽c⩽0.6. Open problems are invoked.