Manipulate[
VectorPlot[(({{a, 1}, {1, a}})).{x,y},{x,-1,1},{y,-1,1},
VectorStyle->Orange,
StreamPoints->Coarse,
StreamScale->Full
],
{a,-2,2}]

RotationMatrix[90°]

StreamPlot[{y,Sin[x]},{x,-4,7},{y,-3,3},
StreamPoints->Coarse
]

contours={-0.8,-0.5,0,0.5,1,1.5,2.2,3};
ei=Show[
ContourPlot[Cos[x]+((y^2)/2),{x,-4,7},{y,-3,3},AspectRatio->1,Axes->True,Frame->False,ContourShading->None,Contours->contours,
Ticks->{Table[{x,""},{x,-4,7,1}]~Join~{-4,-Pi,Pi,2Pi,7},Table[y,{y,-3,3,1}]}],
Graphics[{
Red,PointSize[Medium],Point@Table[{x,0.0},{x,-Pi,2Pi,Pi}]
}],
AspectRatio->0.5]

ei2=Show[ei,
Graphics[{
Arrowheads[.02],
Table[
{
Arrow[{{x,(Sqrt[2(z+1)])},{x+0.1,(Sqrt[2(z+1)])}}],
Arrow[{{x,-(Sqrt[2(z+1)])},{x-0.1,-(Sqrt[2(z+1)])}}]
},
{z,contours},
{x,{-Pi,Pi}}]
}]
]

Export["A3-Erstes-Integral.pdf",ei2]

Show[
ei,
StreamPlot[{y,Sin[x]},{x,-4,7},{y,-3,3},
StreamPoints->{
{0,1},{0,2},{0,1.5},
{0,-1},{0,-2},{0,-1.5},
{-Pi,2},{Pi,2},
{Pi,1},{Pi,1.5}}
]]

With[{a=2},
g=StreamPlot[(({{a, 1}, {1, a}})).{x,y},{x,-1,1},{y,-1,1},
StreamPoints->Coarse,
StreamStyle->Arrowheads[{{0.07,Automatic}}],
StreamScale->Full,
Frame->False,
Axes->True,
Ticks->None,
ImageSize->100
]]

Export["A3-a=2.pdf",g]