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The critical behaviour of $d$-dimensional semi-infinite systems with$n$-component order parameter $\bm{\phi}$ is studied at an $m$-axial bulkLifshitz point whose wave-vector instability is isotropic in an $m$-dimensionalsubspace of $\mathbb{R}^d$. Field-theoretic renormalization group methods areutilised to examine the special surface transition in the case where the $m$potential modulation axes, with $0\leq m\leq d-1$, are parallel to the surface.The resulting scaling laws for the surface critical indices are given. Thesurface critical exponent $\eta_\|^{\rm sp}$, the surface crossover exponent$\Phi$ and related ones are determined to first order in$\epsilon=4+\case{m}{2}-d$. Unlike the bulk critical exponents and the surfacecritical exponents of the ordinary transition, $\Phi$ is $m$-dependent alreadyat first order in $\epsilon$. The $\Or(\epsilon)$ term of $\eta_\|^{\rm sp}$ isfound to vanish, which implies that the difference of $\beta_1^{\rm sp}$ andthe bulk exponent $\beta$ is of order $\epsilon^2$.Comment: 21 pages, one figure included as eps file, uses IOP style file

Boundary critical behaviour atm-axial Lifshitz points: the special transition for the case of a surface plane parallel to the modulation axes