Minimal surfaces in R⁴ foliated by conic sections and parabolic rotations of holomorphic null curves in C⁴

Using the complex parabolic rotations of holomorphic null curves in C⁴, we transform minimal surfaces in Euclidean space R³ ⊂ R⁴ to a family of degenerate minimal surfaces in Euclidean space R⁴. Applying our deformation to holomorphic null curves in C³ ⊂ C⁴ induced by helicoids in R³, we discover new minimal surfaces in R⁴ foliated by conic sections with eccentricity grater than $1$: hyperbolas or straight lines. Applying our deformation to holomorphic null curves in C³ induced by catenoids in R³, we can rediscover the Hoffman-Osserman catenoids in R⁴ foliated by conic sections with eccentricity smaller than $1$: ellipses or circles. We prove the existence of minimal surfaces in R⁴ foliated by ellipses, which converge to circles at infinity. We construct minimal surfaces in R⁴ foliated by parabolas: conic sections which have eccentricity $1$.