R.<u,v,x,y,z> = PolynomialRing(QQ, order='lex')

p = (x-u)^2 + (y-v)^2 + z^2 - 1/10
dpdt = -v*diff(p, u) + u*diff(p, v)

time = u^4 + v^2 - 1

I = ideal(p, dpdt, time)
G = I.groebner_basis()
eqn = G[-1]

print(eqn)

# plotting

var('x0 y0')
eqn = eval(str(eqn).replace('^', '**').replace('x', 'x0').replace('y', 'y0'))
implicit_plot3d(eqn, (-2, 2), (-2, 2), (-2, 2), color='green').show()