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In the framework of k_T-factorization QCD approach and the colour singletmodel we consider J/\psi inelastic photo- and leptoproduction processes atHERA. We investigate the dependences of the single differential and doubledifferential cross section on different forms of the unintegrated gluondistribution. The z and p_T dependences of the spin aligment parameter \alphaare presented also. Our theoretical predictions agree well with recent datataken by the H1 and ZEUS collaborations at HERA. It is shown that experimentalstudy of the polarization J/\psi mesons at low Q^2 < 1 GeV^2 is an additionaltest of BFKL gluon dynamics.Comment: 14 pages, 10 figure

$J/\psi$ production at HERA in the color singlet model with $k_{\mathrm{T}}$ -factorization