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We discuss the prompt production of $J/\psi$ mesons in proton-protoncollisions at the LHC within a NRQCD $k_{t}$-factorization approach usingKimber-Martin-Ryskin (KMR) unintegrated gluon distributions (UGDF). We includeboth direct color-singlet production ($gg \to J/\psi$g) as well as a feed-downfrom $\chi_{c} \to J/\psi \gamma$ and $\psi' \to J/\psi X$. The production ofthe decaying mesons ($\chi_{c}$ or $\psi'$)is also calculated within NRQCD$k_{t}$-factorization. The corresponding matrix elements for $gg \to J/\psi$,$gg \to \psi'$ and $gg \to \chi_{c}$ include parameters of the nonrelativisticspatial wave functions of quarkonia at $r=0$, which are taken from potentialmodels from the literature. We get the ratio of the corresponding of the crosssections for $\chi_{c}(2)$-to-$\chi_{c}(1)$ much closer to experimental datathan obtained in recent analyses. Differential distributions in rapidity of$J/\psi$ and $\psi'$ are calculated and compared to experimental data of theALICE and LHCb collaborations. We discuss possible onset of gluon saturationeffects at forward/backward rapidities. One can describe the experimental datafor $J/\psi$ production within model uncertainties with color-singlet componentonly. Therefore our theoretical results leave only a relatively small room forthe color-octet contributions.

Inclusive production of J/$\psi$ and $\psi'$ mesons at the LHC