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We prove in this short report the existence of a fundamental solution (F.S.)for the Cauchy initial boundary problem on the whole space for the parabolicdifferential equation having at origin the point of non-integrable unboundeddiscontinuity for coefficient before a first order derivative.  We give also the non-asymptotic rapidly decreasing at infinity estimate forthese function.  We extend the classical parametrix method offered by E.E.Levi.

Extremal case of parabolic differential equations having discontinuous  unbounded coefficients. Existence of fundamental solution for an initial  Cauchy problem. Parametrix method