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Using the complex parabolic rotations of holomorphic null curves in${\mathbb{C}}^{4}$, we transform minimal surfaces in Euclidean space${\mathbb{R}}^{3} \subset {\mathbb{R}}^{4}$ to a family of degenerate minimalsurfaces in Euclidean space ${\mathbb{R}}^{4}$. Applying our deformation toholomorphic null curves in ${\mathbb{C}}^{3} \subset {\mathbb{C}}^{4}$ inducedby helicoids in ${\mathbb{R}}^{3}$, we discover new minimal surfaces in${\mathbb{R}}^{4}$ foliated by conic sections with eccentricity grater than$1$: hyperbolas or straight lines. Applying our deformation to holomorphic nullcurves in ${\mathbb{C}}^{3}$ induced by catenoids in ${\mathbb{R}}^{3}$, we canrediscover the Hoffman-Osserman catenoids in ${\mathbb{R}}^{4}$ foliated byconic sections with eccentricity smaller than $1$: ellipses or circles. Weprove the existence of minimal surfaces in ${\mathbb{R}}^{4}$ foliated byellipses, which converge to circles at infinity. We construct minimal surfacesin ${\mathbb{R}}^{4}$ foliated by parabolas: conic sections which haveeccentricity $1$.

Minimal surfaces in ${\mathbb{R}}^{4}$ foliated by conic sections and parabolic rotations of holomorphic null curves in ${\mathbb{C}}^{4}$