We develop solutions for the surface deformation field due to the pressurization of a finite (triaxial) ellipsoidal cavity in a half-space. The solution is in the form of a non-uniform distribution of triaxial point sources within the cavity. The point sources have the same aspect ratio, determined by the cavity shape, while their strengths and spacing are determined in an adaptive manner, such that the net point-source potency per unit volume is uniform. We validate and compare our solution with available analytical and numerical solutions. We provide computationally-efficient MATLAB codes tailored for source inversions. This solution opens the possibility of exploring potential deviations of magma chambers from axi-symmetric geometries.