We complete a fiberwise isotopy classification of smooth real algebraic
and pseudoholomorphic curves of degree 8 on the quadratic cone, which
have a specially shaped oval crossing a given generating line
of the cone in four real points. We link this classification with an
isotopy classification of smoothing of real plane curve singularity
which is the union of
four smooth real local branches quadratically tangent to each other.
(the singularity $X_{21}$).