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We study a self-organized critical system coupled to an isotropic random fluid environment. The former is described by a strongly anisotropic continuous (coarse-grained) model introduced by Hwa and Kardar [Phys. Rev. Lett. {\bf 62} 1813 (1989); Phys. Rev. A {\bf 45} 7002 (1992)]; the latter is described by the stirred Navier--Stokes equation due to Forster, Nelson and Stephen [Phys. Rev. A {\bf 16} 732 (1977)]. The full problem of two coupled stochastic equations is represented as a field theoretic model, which is shown to be multiplicatively renormalizable. The corresponding renormalization group equations possess a semi-infinite curve of fixed points in the four-dimensional space of the model parameters. The whole curve is infrared attractive for realistic values of parameters; its endpoint corresponds to the purely isotropic regime where the original Hwa-Kardar nonlinearity becomes irrelevant. There, one is left with a simple advection of a passive scalar field by the external environment. The main critical dimensions are calculated to the leading one-loop order (first terms in the $\varepsilon=4-d$ expansion); some of them are appear to be exact in all orders. They remain the same along that curve, which makes it reasonable to interpret it as a single universality class. However, the correction exponents do vary along the curve. It is therefore not clear whether the curve survives in all orders of the renormalization group expansion or shrinks to a single point when the higher-order corrections are taken into account.