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A generalized Bessel function of two variables satisfies a system of partial differential equations. Two of the singular loci of the system are of irregular type. Near one of them we study the asymptotic behavior of suitably chosen linearly independent solutions. In our calculation coalescing saddle points are treated. Introduction. Let z* be a function of (x, y)eC defined by (0.1) Jc*_ )c where α (φZ) is a complex constant and C*=(t: oo—> oo; argt: —π—•π) is a loop starting from t= oo, encircling / = 0 in the positive sense and returning to t= oo along which aτgt varies from — π to π. It is known that (0.1) is a solution of a system of partial differential equations dχiι = xdxu —ydyu — ecu , (0.2) dxdyu = u, ydyU = — dxu — (α + 1 )dyu + xu (see [6]), which is equivalent to a completely integrable Pfaffian system of the form (0.3) dV= (P(x, y)dx + Q(x, y)dy/y) V with / 0 1 0 \ / 0 0 y P(χ,y)=l -« x -y ) , Q(χ,y)=l y o o \ 1 0 0 / \ x 1 ( α + l ) System (0.2) or (0.3) possesses the singular loci x=oo, y=co of irregular type, and y = 0 of regular type. For a fixed point (xo,yo)eCxC x (C =C— {0}), the 1991 Mathematics Subject Classification. Primary 33C70; Secondary 35C20.

On a generalized Bessel function of two variables, II. Case of coalescing saddle points