Consider a Hamiltonian of the form \[ H=T+V,\qquad T={1\over 2}\sum^ n_{i,j=1}a_{i,j}y_ iy_ j,\qquad V=\sum_{m\in Z^ n}v_ m\exp(m,x) \] where \((a_{i,j})\) is a nondegenerate constant matrix \(v_ m=\text{const}\), \(x=(x_ 1,\dots,x_ n)\) and \(y=(y_ 1,\dots,y_ n)\) are conjugate canonical variables. The author shows that if a system with the above Hamiltonian has \(r>0\) first integrals functionally independent of \(H\), which are polynomials with respect to \(y\) with the coefficients of the form \(\sum_{m\in Z^ n}f_ m\exp(m,x)\), then it has \(r\) first integrals of the same form which are functionally independent of \(H\) in the highest terms. The paper may be considered as an appendix to the paper by \textit{V. V. Kozlov} and \textit{D. V. Treshchev} [Math. USSR, Izv. 34, No. 3, 555-574 (1990); translation from Izv. Akad. Nauk SSSR, Ser. Mat. No. 3, 537-556 (1989; Zbl 0684.58012)] where several results on the absence of the above integrals, functionally independent of \(H\) in the highest terms, for systems with the above \(H\) were obtained.