Mixed Simultaneous Cyclicity for Piecewise Smooth Vector Fields Defined on an Invariant Sphere

In this work we consider the class of partially integrable 3-dimensional piecewise smooth vector fields Y=(X⁺,X⁻) Y = ( X + , X - ) with separation set Σ =\(x,y,z)∈ R³: z=0\ Σ = { ( x , y , z ) ∈ R 3 : z = 0 } and first integral H(x,y,z)=x²+y²+z² H ( x , y , z ) = x 2 + y 2 + z 2 that leaves invariant any sphere centered at the origin, S_ρ ²=\(x,y,z)∈ R³: x²+y²+z²=ρ ²\ S ρ 2 = { ( x , y , z ) ∈ R 3 : x 2 + y 2 + z 2 = ρ 2 } . We denote this class by X X and by Xₙ X n when X^± X ± are polynomial vector fields of degree n . Our main goal is to study piecewise smooth vector fields in the class X₁ X 1 presenting three periodic annuli on the invariant sphere S₁² S 1 2 </mml:msubsup