  The isoscalar proton-neutron pairing and isovector pairing, including both
isovector proton-neutron pairing and like-particle pairing, are treated in a
formalism which conserves exactly the particle number and the isospin. The
formalism is designed for self-conjugate (N=Z) systems of nucleons moving in
axially deformed mean fields and interacting through the most general isovector
and isoscalar pairing interactions. The ground state of these systems is
described by a superposition of two types of condensates, i.e., condensates of
isovector quartets, built by two isovector pairs coupled to the total isospin
T=0, and condensates of isoscalar proton-neutron pairs. The comparison with the
exact solutions of realistic isovector-isoscalar pairing Hamiltonians shows
that this ansatz for the ground state is able to describe with high precision
the pairing correlation energies. It is also shown that, at variance with the
majority of Hartree-Fock-Bogoliubov calculations, in the present formalism the
isovector and isoscalar pairing correlations coexist for any pairing
interactions. The competition between the isovector and isoscalar
proton-neutron pairing correlations is studied for N=Z nuclei with the valence
nucleons moving in the $sd$ and $pf$ shells and in the major shell above
$^{100}$Sn. We find that in these nuclei the isovector pairing prevail over the
isoscalar pairing, especially for heavier nuclei. However, the isoscalar
proton-neutron correlations are significant in all nuclei and they always
coexist with the isovector pairing correlations.
