  We investigate the construction of the quantum commuting hamiltonians for the
Gaudin integrable model. We prove that [Tr L^k(z), Tr L^m(u) ]=0, for k,m < 4 .
However this naive receipt of quantization of classically commuting
hamiltonians fails in general, for example we prove that [Tr L^4(z), Tr L^2(u)
] \ne 0. We investigate in details the case of the one spin Gaudin model with
the magnetic field also known as the model obtained by the "argument shift
method". Mathematically speaking this method gives maximal Poisson commutative
subalgebras in the symmetric algebra S(gl(N)). We show that such subalgebras
can be lifted to U(gl(N)), simply considering Tr L(z)^k, k\le N for N<5. For
N=6 this method fails: [Tr L_{MF}(z)^6, L_{MF}(u)^3]\ne 0 . All the proofs are
based on the explicit calculations using r-matrix technique. We also propose
the general receipt to find the commutation formula for powers of Lax operator.
For small power exponents we find the complete commutation relations between
powers of Lax operators.
