Central force and the higher rank numerical range

Let r = r ( θ ) be the orbit of a point mass under a central force f ( r ) = − 1 / r 3 with angular momentum M. Suppose p = M / M 2 − 1 . We show that the orbit is a transcendental curve if p is irrational, and the orbit is an algebraic curve F A ( 1 , x , y ) = 0 for some 2 m × 2 m nilpotent Toeplitz matrix A if p = m / j 0 is rational, where F ( t , x , y ) = F A ( t , x , y ) = det ( t I n + x ( A + A ⁎ ) / 2 + y ( A − A ⁎ ) / ( 2 i ) ) . Furthermore, we examine the rank-k numerical range Λ k ( A ) of this nilpotent Toeplitz matrix, showing that the sum of numbers of flat portions on the boundary of Λ k ( A ) , k = 1 , 2 , … , m , is ( m − 1 ) ( 2 m − 3 ) .