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We analyse the combined effect of orbital and Pauli depairing on thesuperconducting state, and apply the results to the heavy fermion CeCoIn$_5$.We find that: a) standard extrapolation based on the slope of $H_{c2}(T)$ inthe vicinity of the transition temperature to T=0 does not always give accuratevalues of the orbital upper critical field; b) critical value of the Makiparameter, $\alpha$, that determines onset of the first order transitiondepends on the Fermi surface shape and the symmetry of the gap, and is$\alpha^*\approx 3$ for CeCoIn$_5$; c) the anisotropy of the thermodynamic andtransport coefficients in the low-temperature, low-field part of the phasediagram is essentially insensitive to the Zeeman field and can be used todetermine the nodal directions in Pauli-limited superconductors. The latterresult confirms the finding of the $d_{x^2-y^2}$ order parameter CeCoIn$_5$.

Vortex state in d-wave superconductors with strong paramagnetism: transport and specific heat anisotropy