  Using first principles calculations based on density functional theory, we
study the geometric, electronic, and magnetic properties of Pt, Ni and Co-based
half Heusler alloys, namely, Pt$BC$, Ni$BC$ and Co$BC$ ($B$ = Cr, Mn and Fe;
$C$ = Al, Si, P, S, Ga, Ge, As, Se, In, Sn, Sb and Te). We calculate the
formation energy of these alloys in various crystal symmetries, which include,
the (face-centered) cubic $C1_{b}$ ($F\bar{4}$3m), orthorhombic ($Pnma$), as
well as hexagonal ($P\bar{6}2m$ and $P6_{3}/mmc$) structures. It has been
observed that out of all the 108 structures, studied here, energetically stable
cubic structure is observed for only 18 materials. These alloys are primarily
having either a $C$ atom or an $A$ atom with a high atomic number. We also
observe that along with the alloys with $C$ atoms from group IIIA, IVA and VA
-- alloys with $C$ atoms from group VIA are also found to be, by and large,
energetically stable. To examine the relative stabilities of different
symmetries in order to search for the respective lowest energy state for each
of the above-mentioned systems, as well as to find whether a material in the
ground state is half-metallic or not, we analyze the formation energy, and the
electronic density of states, in detail. Based on these analyses, the
possibility of existence of any {\it one-to-one relationship} between the {\it
cubic symmetry} and the {\it half-metallicity} in these half Heusler alloys is
probed. Subsequently, we predict about the existence of a few new {\it
non-cubic} half Heusler alloys with substantially low density of states at one
of the spin channels and reasonably {\it high spin polarization at the Fermi
level}
