Optimal designs for regression with asymmetric errors build on the second-order least squares estimator (SLSE) of Wang and Leblanc (2008), whose gain over ordinary least squares (OLS) comes from the error skewness. With i.i.d. errors the optimally weighted SLSE solves the same estimating equation as the degree-two Polynomial Maximization Method (PMM), a family whose degree-$S$ location estimator has asymptotic variance $g_S\le1$ times that of the sample mean. We derive optimal designs for this family when the errors are independent with constant variance but have standardised cumulants of order three and higher that vary with the regressor (cumulant heteroscedasticity). For a known reduction field $g_S(x)$, the $g_S$-weighted information matrix $M_{g_S}$ obeys the Kiefer-Wolfowitz equivalence theorem. A constant field leaves the OLS-optimal design unchanged; a non-constant field can change it but need not. For symmetric errors with varying kurtosis, the PMM$_2$ and conditional-SLSE designs cannot depart from the OLS-optimal one, whereas a degree-three design can. A stratify-and-pool PMM$_S$ estimator attains $N\mathrm{Cov}(\hatθ)\toσ^2M_{g_S}^{-1}$ at every degree as all strata grow; its per-stratum reduction is below one only for symmetric errors with the symmetry imposed. For a quadratic model on $[0,1]$ with symmetric Beta errors of unit variance and dose-varying shape, the third-order design for the known field has $A$-efficiency $1.047$ relative to the OLS-optimal design, and $1.001$ under the $D$-criterion. A two-stage procedure that estimates the field from a one-third pilot builds designs of $A$-efficiency $1.014$-$1.020$ at $N=600$-$2400$; its realised mean-squared-error ratio against the OLS-optimal design ranges from $0.89$ (a loss) to $1.08$.

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