  While it has been pointed out that the chiral symmetry, which is important
for the Dirac fermions in graphene, can be generalized to tilted Dirac fermions
as in organic metals, such a generalized symmetry was so far defined only for a
continuous low-energy Hamiltonian. Here we show that the generalized chiral
symmetry can be rigorously defined for lattice fermions as well. A key concept
is a continuous "algebraic deformation" of Hamiltonians, which generates
lattice models with the generalized chiral symmetry from those with the
conventional chiral symmetry. This enables us to explicitly express zero modes
of the deformed Hamiltonian in terms of that of the original Hamiltonian.
Another virtue is that the deformation can be extended to non-uniform systems,
such as fermion-vortex systems and disordered systems. Application to fermion
vortices in a deformed system shows how the zero modes for the conventional
Dirac fermions with vortices can be extended to the tilted case.
