Waring's problem for a ternary quadratic form and an arbitrary even power

One obtains asymptotic formulas for the number of solutions of the equation $n=f(x, y, z)+w^{2k}$, where $f$ is a primitive integral quadratic form. One gives an estimate of the remainder, having a logarithmic reducing factor in the general case and a powerlike one when $f(x ,y, z)=x^2+y^2+z^2$.