Thermodynamic properties, $^{31}$P nuclear magnetic resonance (NMR) measurements, and density-functional band-structure calculations for $varepsilon$-LiVOPO$_4$ are reported. This quantum magnet features a singlet ground state and comprises two types of alternating spin-$frac12$ chains that manifest themselves by the double maxima in the susceptibility and magnetic specific heat, and by the two-step magnetization process with an intermediate $frac12$-plateau. From thermodynamic data and band-structure calculations, we estimate the leading couplings of $J_1simeq 20$ K and $J_2simeq 60$ K and the alternation ratios of $alpha_1=J_1/J_1simeq 0.6$ and $alpha_2=J_2/J_2simeq 0.3$ within the two chains, respectively. The zero-field spin gap $Delta_0/k_{rm B}simeq 7.3$ K probed by thermodynamic and NMR measurements is caused by the $J_1$-$J_1$ spin chains and can be closed in the applied field of $mu_{0}H_{rm c1}simeq 5.6$ T, giving rise to a field-induced long-range order. The NMR data reveal predominant three-dimensional spin-spin correlations at low temperatures. Field-induced magnetic ordering transition observed above $H_{c1}$ is attributed to the Bose-Einstein condensation of triplons in the sublattice formed by the $J_1$-$J_1$ chains with weaker exchange couplings.
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