In this paper we introduce an exactly solvable Kondo lattice model without any fine-tuning local gauge symmetry. This model describes itinerant electrons interplaying with a localized magnetic moment via only longitudinal Kondo exchange. Its solvability results from conservation of the localized moment at each site, and is valid for arbitrary lattice geometry and electron filling. A case study on square lattice shows that the ground state is a N{e}el antiferromagnetic insulator at half-filling. At finite temperature, paramagnetic phases including a Mott insulator and correlated metal are found. The former is a melting antiferromagnetic insulator with a strong short-range magnetic fluctuation, while the latter corresponds to a Fermi liquid-like metal. Monte Carlo simulation and theoretical analysis demonstrate that the transition from paramagnetic phases into the antiferromagnetic insulator is a continuous $2D$ Ising transition. Away from half-filling, patterns of spin stripes (inhomogeneous magnetic order) at weak coupling, and phase separation at strong coupling are predicted. With established Ising antiferromagnetism and spin stripe orders, our model may be relevant to a heavy fermion compound CeCo(In$_{1-x}$Hg$_{x}$)$_{5}$ and novel quantum liquid-crystal order in a hidden order compound URu$_{2}$Si$_{2}$.
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