BH01 Light from Schwarzschild black holes (I.I. Cotaescu) Conserved quantities in BH comoving frame E0,P01,P02,... Conserved quantities in Observer's frame: E,P1,P2,.... Final results after fixing the angle 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 omega[H]=omega LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEocmVzdGFydEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIjtGJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRjEvJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdRLDAuMjc3Nzc3OGVtRic= QyQtSSdhc3N1bWVHNiI2JS1JIj5HJSpwcm90ZWN0ZWRHNiRJIlZHRiUiIiEtRig2JEkmZGVsdGFHRiVGLC1GKDYkSSZvbWVnYUdGJUYsIiIi metric5:=<<1|0|0|0|0>,<0|-1|0|0|0>,<0|0|-1|0|0>,<0|0|0|-1|0>,<0|0|0|0|-1>>: Id_5:=<<1|0|0|0|0>,<0|1|0|0|0>,<0|0|1|0|0>,<0|0|0|1|0>,<0|0|0|0|1>>: eta:=<<1|0|0|0>,<0|-1|0|0>,<0|0|-1|0>,<0|0|0|-1>>: Id:=<<1|0|0|0>,<0|1|0|0>,<0|0|1|0>,<0|0|0|1>>: QyY+SSJkRzYiKiZJJmRlbHRhR0YlIiIiSSZvbWVnYUdGJSEiIkYoPkkiUkdGJSomRiRGKEkjeGlHRiVGKEYo

KiZJJ2RlbHRhfGlyRzYiIiIiSSdvbWVnYXxpckdGJCEiIg==

KihJJ2RlbHRhfGlyRzYiIiIiSSdvbWVnYXxpckdGJCEiIkkjeGlHRiRGJQ==

Translations generated by LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUklbXN1YkdGJDYlLUkjbWlHRiQ2JlEiUEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZm9yZWdyb3VuZEdRKlswLDAsMjU1XUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYmLUYvNiZRImlGJ0YyRjVGOEYyRjVGOC8lL3N1YnNjcmlwdHNoaWZ0R1EiMEYnRjJGNUY4 P_mat:=proc(x,y,z) local mat; mat:= I*omega*(<<0|x|y|z|0>,<x|0|0|0|x>,<y|0|0|0|y>,<z|0|0|0|z>,<0|-x|-y|-z|0>>); mat;end proc: Trans_P:=proc(x,y,z) global Id_5,P_mat; return evalm(Id_5-I*P_mat(x,y,z)-(1/2)*(P_mat(x,y,z).P_mat(x,y,z)));end proc: QyQ+SSVUcmFuRzYiLUkoVHJhbnNfUEdGJTYlSSJkR0YlIiIhRioiIiI=

PTYiNiQ7IiIiIiImRiVFXFtsOjYkIiIkRipGJjYkRipGJiIiITYkRiZGKkYsNiRGJyIiJUYsNiRGL0YvRiY2JCIiI0YyRiY2JEYnRicsJkYmRiYqJEknZGVsdGF8aXJHRiNGMiMhIiJGMjYkRidGJiwkRjVGNzYkRi9GKkYsNiRGJ0YqRiw2JEYmRiYsJkYmRiZGNSNGJkYyNiRGJkYvRiw2JEYyRi9GLDYkRi9GJkYsNiRGKkYyRiw2JEYqRi9GLDYkRjJGKkYsNiRGJkYyRjY2JEYvRidGLDYkRjJGJkY2NiRGL0YyRiw2JEYmRicsJEY1Rj82JEYqRidGLDYkRidGMiwkRjZGODYkRjJGJ0Y2

Boosts generated by LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUklbXN1YkdGJDYlLUkjbWlHRiQ2JlEiS0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZm9yZWdyb3VuZEdRKlswLDAsMjU1XUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYmLUYvNiZRImlGJ0YyRjVGOEYyRjVGOC8lL3N1YnNjcmlwdHNoaWZ0R1EiMEYnRjJGNUY4 K_mat:=proc(x,y,z) return matrix(5,5,[[0,x,y,z,0],[x,0,0,0,0],[y,0,0,0,0],[z,0,0,0,0],[0,0,0,0,0]]); end proc: Boost_K:=proc(x,y,z) global Id_5, K_mat; local r,A; r:=sqrt(x^2+y^2+z^2);A:=evalm(K_mat(x,y,z)/r);return simplify(evalm(Id_5 + A.A*(cosh(r)-1)+sinh(r)*A),symbolic);end proc: QyQ+SSViZXRhRzYiLUkoYXJjdGFuaEdGJTYjSSJWR0YlIiIi

LUkoYXJjdGFuaEc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYjSSNWfGlyR0Yn

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PTYiNiQ7IiIiIiImRiVFXFtsOjYkIiIkRipGJjYkRipGJiIiITYkRiZGKkYsNiRGJyIiJUYsNiRGL0YvRiY2JCIiI0YyKiQsJiokSSNWfGlyR0YjRjIhIiJGJkYmI0Y3RjI2JEYnRidGJjYkRidGJkYsNiRGL0YqRiw2JEYnRipGLDYkRiZGJkYzNiRGJkYvRiw2JEYyRi9GLDYkRi9GJkYsNiRGKkYyRiw2JEYqRi9GLDYkRjJGKkYsNiRGJkYyKiZGNkYmRjRGODYkRi9GJ0YsNiRGMkYmRkU2JEYvRjJGLDYkRiZGJ0YsNiRGKkYnRiw2JEYnRjJGLDYkRjJGJ0Ys

Initial conserved quantities in O_BH Eqs. (38)-(41) QyQ+SSNFMEc2Ikkia0dGJSIiIg==

SSJrRzYi

Qyg+SSRQMDFHNiIsJComSSJrR0YlIiIiLUkkY29zRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YlNiNJJmFscGhhR0YlRikhIiJGKT5JJFAwMkdGJSwkKiZGKEYpLUkkc2luR0YsRi9GKUYxRik+SSRQMDNHRiUiIiFGKQ==

LCQqJkkia0c2IiIiIi1JJGNvc0c2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJTYjSSZhbHBoYUdGJUYmISIi

LCQqJkkia0c2IiIiIi1JJHNpbkc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJTYjSSZhbHBoYUdGJUYmISIi

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Qyg+SSRRMDFHNiIqJkkia0dGJSIiIiwoKigtSSRjb3NHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiU2I0kmYWxwaGFHRiVGKEkmZGVsdGFHRiUiIiNJI3hpR0YlRjNGKCooLUkkc2luR0YtRjBGKEYyRihGNEYoISIjRishIiJGKEYoPkkkUTAyR0YlKiZGJ0YoLCgqKEY2RihGMkYzRjRGM0YoKihGK0YoRjJGKEY0RihGM0Y2RjlGKEYoPkkkUTAzR0YlIiIhRig=

KiZJImtHNiIiIiIsKCooLUkkY29zRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiNJJmFscGhhR0YkRiVJJ2RlbHRhfGlyR0YkIiIjSSN4aUdGJEYwRiUqKC1JJHNpbkdGKkYtRiVGL0YlRjFGJSEiI0YoISIiRiU=

KiZJImtHNiIiIiIsKCooLUkkc2luRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiNJJmFscGhhR0YkRiVJJ2RlbHRhfGlyR0YkIiIjSSN4aUdGJEYwRiUqKC1JJGNvc0dGKkYtRiVGL0YlRjFGJUYwRighIiJGJQ==

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Qyg+SSRMMDNHNiIqKkkmZGVsdGFHRiUiIiJJImtHRiVGKEkjeGlHRiVGKEkmb21lZ2FHRiUhIiJGKD5JJEwwMUdGJSIiIUYoPkkkTDAyR0YlRi9GKA==

KipJJ2RlbHRhfGlyRzYiIiIiSSJrR0YkRiVJI3hpR0YkRiVJJ29tZWdhfGlyR0YkISIi

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Invariant (30) LUkpc2ltcGxpZnlHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHNiI2IywsKiRJI0UwR0YnIiIjIiIiKiZJJm9tZWdhR0YnRiwsKCokSSRMMDFHRidGLEYtKiRJJEwwMkdGJ0YsRi0qJEkkTDAzR0YnRixGLUYtISIiKiZJJFAwMUdGJ0YtSSRRMDFHRidGLUY3KiZJJFAwMkdGJ0YtSSRRMDJHRidGLUY3KiZJJFAwM0dGJ0YtSSRRMDNHRidGLUY3

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K , R and L components of the tensor Gen as in Eqs.(31). Notation (32) K(x,P)=Gen Qyg+SSRLMDFHNiIqJiwmSSRRMDFHRiUiIiJJJFAwMUdGJSEiIkYpLUkiL0c2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJTYjLCRJJm9tZWdhR0YlIiIjRilGKT5JJEswMkdGJSomLCZJJFEwMkdGJUYpSSRQMDJHRiVGK0YpRixGKUYpPkkkSzAzR0YlKiYsJkkkUTAzR0YlRilJJFAwM0dGJUYrRilGLEYpRik=

LCQqJiwmKiZJImtHNiIiIiIsKCooLUkkY29zRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YnNiNJJmFscGhhR0YnRihJJ2RlbHRhfGlyR0YnIiIjSSN4aUdGJ0YzRigqKC1JJHNpbkdGLUYwRihGMkYoRjRGKCEiI0YrISIiRihGKComRiZGKEYrRihGKEYoSSdvbWVnYXxpckdGJ0Y5I0YoRjM=

LCQqJiwmKiZJImtHNiIiIiIsKCooLUkkc2luRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YnNiNJJmFscGhhR0YnRihJJ2RlbHRhfGlyR0YnIiIjSSN4aUdGJ0YzRigqKC1JJGNvc0dGLUYwRihGMkYoRjRGKEYzRishIiJGKEYoKiZGJkYoRitGKEYoRihJJ29tZWdhfGlyR0YnRjgjRihGMw==

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Qyg+SSRSMDFHNiIqJiwmSSRRMDFHRiUhIiJJJFAwMUdGJUYpIiIiLUkiL0c2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJTYjLCRJJm9tZWdhR0YlIiIjRitGKz5JJFIwMkdGJSomLCZJJFEwMkdGJUYpSSRQMDJHRiVGKUYrRixGK0YrPkkkUjAzR0YlKiYsJkkkUTAzR0YlRilJJFAwM0dGJUYpRitGLEYrRis=

LCQqJiwmKiZJImtHNiIiIiIsKCooLUkkY29zRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YnNiNJJmFscGhhR0YnRihJJ2RlbHRhfGlyR0YnIiIjSSN4aUdGJ0YzRigqKC1JJHNpbkdGLUYwRihGMkYoRjRGKCEiI0YrISIiRihGOSomRiZGKEYrRihGKEYoSSdvbWVnYXxpckdGJ0Y5I0YoRjM=

LCQqJiwmKiZJImtHNiIiIiIsKCooLUkkc2luRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YnNiNJJmFscGhhR0YnRihJJ2RlbHRhfGlyR0YnIiIjSSN4aUdGJ0YzRigqKC1JJGNvc0dGLUYwRihGMkYoRjRGKEYzRishIiJGKEY4KiZGJkYoRitGKEYoRihJJ29tZWdhfGlyR0YnRjgjRihGMw==

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QyQ+SSRHZW5HNiItSSdtYXRyaXhHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiU2JSIiJkYsNyc3JyIiISomSSZvbWVnYUdGJSIiIkkkSzAxR0YlRjIqJkYxRjJJJEswMkdGJUYyKiZGMUYySSRLMDNHRiVGMkkjRTBHRiU3JywkRjAhIiJGLyomRjFGMkkkTDAzR0YlRjIsJComRjFGMkkkTDAyR0YlRjJGOyomRjFGMkkkUjAxR0YlRjI3JywkRjRGOywkRjxGO0YvKiZGMUYySSRMMDFHRiVGMiomRjFGMkkkUjAyR0YlRjI3JywkRjZGO0Y/LCRGRkY7Ri8qJkYxRjJJJFIwM0dGJUYyNycsJEY4RjssJEZBRjssJEZIRjssJEZNRjtGL0Y7 SO(1,4) matrix transforming Gen-> GEN. The components of GEN are the conserved quantities in O TTB:=simplify(evalm(metric5.Lor.Tran.metric5));

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GEN:=simplify(evalm(TTB.Gen.transpose(TTB))): 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 E:=simplify(GEN[1,5],symbolic);

LCQqKEkia0c2IiIiIiwwKiotSSRjb3NHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiU2I0kmYWxwaGFHRiVGJkkjVnxpckdGJUYmSSdkZWx0YXxpckdGJSIiI0kjeGlHRiVGMkYmKihGKUYmRjBGJkYxRjIhIiIqKi1JJHNpbkdGK0YuRiZGMEYmRjFGJkYzRiYhIiMqJkYpRiZGMEYmRjkqJkYpRiZGMUYmRjkqJkYwRiZGMUYmRjJGMkYmRiYsJiokRjBGMkY1RiZGJiNGNUYyI0YmRjI=

P1:=simplify(-(GEN[1,2]+GEN[2,5]),symbolic);

LCQqKEkia0c2IiIiIiw4KiotSSRjb3NHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiU2I0kmYWxwaGFHRiVGJiwmKiRJI1Z8aXJHRiUiIiMhIiJGJkYmI0YmRjNJJ2RlbHRhfGlyR0YlRjNJI3hpR0YlRjNGJiooRilGJkY2RjNGN0YzRjQqKEYpRiZGMEY1RjZGM0Y0KiotSSRzaW5HRitGLkYmRjBGNUY2RiZGN0YmISIjKihGKUYmRjJGJkY2RiZGMyomRjZGM0YpRiZGJiooRjtGJkY2RiZGN0YmRjMqJkY2RiZGMEY1RjNGKUYzRjJGPUY2Rj1GJkYwI0Y0RjNGQg==

P2:=simplify(-(GEN[1,3]+GEN[3,5]));

LCQqKEkia0c2IiIiIiw4KiotSSRzaW5HNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiU2I0kmYWxwaGFHRiVGJiwmKiRJI1Z8aXJHRiUiIiMhIiJGJkYmI0YmRjNJJ2RlbHRhfGlyR0YlRjNJI3hpR0YlRjNGJiooRilGJkY2RjNGN0YzRjQqKi1JJGNvc0dGK0YuRiZGMEY1RjZGJkY3RiZGMyooRilGJkYwRjVGNkYzRiYqKEYwRjVGNkYzRjdGJiEiIyooRjpGJkY2RiZGN0YmRj4qKEYpRiZGMkYmRjZGJkY+KiZGNkYzRilGJkY0KihGMkYmRjZGJkY3RiZGMyomRjZGM0Y3RiZGMyomRilGJkYwRjVGPkYmRjAjRjRGM0Y1

P3:=simplify(-(GEN[1,4]+GEN[4,5]));

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Q1:=simplify(-(GEN[2,5]-GEN[1,2]));

LCQqKEkia0c2IiIiIiw4KiotSSRjb3NHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiU2I0kmYWxwaGFHRiVGJiwmKiRJI1Z8aXJHRiUiIiMhIiJGJkYmI0YmRjNJJ2RlbHRhfGlyR0YlRjNJI3hpR0YlRjNGJiooRilGJkY2RjNGN0YzRiYqKEYpRiZGMEY1RjZGM0Y0KiotSSRzaW5HRitGLkYmRjBGNUY2RiZGN0YmISIjKihGKUYmRjJGJkY2RiZGPSomRjZGM0YpRiZGNCooRjtGJkY2RiZGN0YmRj0qJkY2RiZGMEY1RjNGKUY9RjJGM0Y2RjNGJkYwI0Y0RjNGNQ==

Q2:=simplify(-(GEN[3,5]-GEN[1,3]));

LCQqKEkia0c2IiIiIiw4KiotSSRzaW5HNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiU2I0kmYWxwaGFHRiVGJiwmKiRJI1Z8aXJHRiUiIiMhIiJGJkYmI0YmRjNJJ2RlbHRhfGlyR0YlRjNJI3hpR0YlRjNGJiooRilGJkY2RjNGN0YzRiYqKi1JJGNvc0dGK0YuRiZGMEY1RjZGJkY3RiZGMyooRilGJkYwRjVGNkYzRiYqKEYwRjVGNkYzRjdGJiEiIyooRjpGJkY2RiZGN0YmRjMqKEYpRiZGMkYmRjZGJkYzKiZGNkYzRilGJkYmKihGMkYmRjZGJkY3RiZGPiomRjZGM0Y3RiZGPiomRilGJkYwRjVGPkYmRjAjRjRGM0Y1

Q3:=simplify(-(GEN[4,5]-GEN[1,4]));

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L3:=collect(simplify(GEN[2,3]/omega),delta);

LCYqLEkia0c2IiIiIiwoKigtSSRzaW5HNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiU2I0kmYWxwaGFHRiVGJkkjVnxpckdGJUYmSSN4aUdGJSIiI0YmKiZGKUYmRjBGJkYmKiZGMUYmRjBGJiEiI0YmLCYqJEYwRjIhIiJGJkYmI0Y4RjJJJ29tZWdhfGlyR0YlRjhJJ2RlbHRhfGlyR0YlRjJGOSosRiRGJiwoKigtSSRjb3NHRitGLkYmRjBGJkYxRiZGMkYpRjJGMUY1RiZGNkY5RjpGOEY7RiZGOQ==

L1:=GEN[3,4]/omega;

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L2:=GEN[4,2]/omega;

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Invariant (30) QyQtSSlzaW1wbGlmeUc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYjLCwqJEkiRUdGKCIiIyIiIiomSSZvbWVnYUdGKEYtLCgqJEkjTDFHRihGLUYuKiRJI0wyR0YoRi1GLiokSSNMM0dGKEYtRi5GLiEiIiomSSNQMUdGKEYuSSNRMUdGKEYuRjgqJkkjUDJHRihGLkkjUTJHRihGLkY4KiZJI1AzR0YoRi5JI1EzR0YoRi5GOEYu

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Solving equation J12=0 with the trick 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 QyQ+SSNFcUc2Ii1JKGNvbGxlY3RHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiU2JC1JJm51bWVyR0YpNiMqJkkjTDNHRiUiIiItSSIvR0YoNiMsJComSSJrR0YlRjFJJm9tZWdhR0YlRjEiIiNGMS1JJHNpbkdGKDYjSSZhbHBoYUdGJUYx

LCYqKkkia0c2IiIiIkknZGVsdGF8aXJHRiVGJiwoKihJI1Z8aXJHRiVGJkYnRiZJI3hpR0YlIiIjRiYqJkYqRiZGJ0YmRiZGLEYmRiYtSSRzaW5HNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiU2I0kmYWxwaGFHRiVGJiEiIiooRiRGJkYnRiYsKCooLUkkY29zR0YwRjNGJkYqRiZGK0YmRiwqKEYqRiZGJ0YmRitGJiEiI0YrRjxGJkY1

QyQ+SSN0ckc2IjwkLy1JJGNvc0c2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJTYjSSZhbHBoYUdGJSomLCYqJEkkdGF1R0YlIiIjISIiIiIiRjVGNSwmRjFGNUY1RjVGNC8tSSRzaW5HRipGLSwkKiZGMkY1RjZGNEYzRjU=

PCQvLUkkY29zRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiNJJmFscGhhR0YpKiYsJiokSSR0YXVHRikiIiMhIiIiIiJGMkYyLCZGLkYyRjJGMkYxLy1JJHNpbkdGJkYqLCQqJkYvRjJGM0YxRjA=

QyQ+SSNlcUc2Ii1JKGNvbGxlY3RHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiU2JC1JJm51bWVyR0YpNiMtSSlzaW1wbGlmeUdGKDYjLUklc3Vic0dGKTYkSSN0ckdGJUkjRXFHRiVJJHRhdUdGJSIiIg==

LCgqKkkia0c2IiIiIkknZGVsdGF8aXJHRiVGJiwoKihJI1Z8aXJHRiVGJkYnRiZJI3hpR0YlRiZGJiomRipGJkYrRiZGJkYrRiZGJkkkdGF1R0YlIiIjRi4qKkYkRiZGJ0YmLCgqKEYqRiZGJ0YmRitGLiEiIiomRipGJkYnRiZGMiEiI0YmRiZGLUYmRi4qKEYkRiZGJ0YmLChGKUYmRixGMkYrRiZGJkYu

QyQ+SSRzb2xHNiItSSZzb2x2ZUdGJTYkSSNlcUdGJUkkdGF1R0YlIiIi

NiQsJCooLCoqKEkjVnxpckc2IiIiIkknZGVsdGF8aXJHRihGKUkjeGlHRigiIiNGKSomRidGKUYqRilGKUYsRikqJCwyKihGJ0YsRipGLEYrIiIlRikqKEYnRixGKkYsRitGLCEiIyomRidGLEYqRixGKSomRidGLEYrRixGMUYmISIlRi1GMSokRitGLEY2RjFGKSNGKUYsRilGKUYrISIiLChGLUYpRidGKUYpRilGOUY4LCQqKCwqRiZGOUYtRjlGLkYpRjNGKUYpRitGOUY6RjkjRjlGLA==

Selecting solutions QyY+SSZzaW5hMUc2Ii1JKXNpbXBsaWZ5RzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YlNiQsJComJkkkc29sR0YlNiMiIiJGMSwmKiRGLiIiI0YxRjFGMSEiIkY0SSlzeW1ib2xpY0dGJUY1LUYnNiQtSSdzZXJpZXNHRik2JUYkL0kjeGlHRiUiIiEiIiRGNkYx

KydJI3hpRzYiLCQqJiwoKiZJI1Z8aXJHRiQiIiJJJ2RlbHRhfGlyR0YkRipGKkYpRipGKkYqRiosJkYoRioiIiNGKiEiIkYtRiotSSJPRyUqcHJvdGVjdGVkRzYjRioiIiQ=

QyY+SSZjb3NhMUc2IiomLCYqJCZJJHNvbEdGJTYjIiIiIiIjISIiRixGLEYsLCZGKEYsRixGLEYuRi4tSSlzaW1wbGlmeUc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJTYkLUknc2VyaWVzR0YzNiVGJC9JI3hpR0YlIiIhIiIkSSlzeW1ib2xpY0dGJUYs

KylJI3hpRzYiISIiIiIhLCQqJiwoKiZJI1Z8aXJHRiQiIiJJJ2RlbHRhfGlyR0YkRixGLEYrRixGLEYsIiIjLCZGKkYsRi5GLCEiI0YuRi4tSSJPRyUqcHJvdGVjdGVkRzYjRiwiIiU=

QyY+SSZzaW5hMkc2Ii1JKXNpbXBsaWZ5RzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YlNiQsJComJkkkc29sR0YlNiMiIiMiIiIsJiokRi5GMUYyRjJGMiEiIkYxSSlzeW1ib2xpY0dGJUY1LUYnNiQtSSdzZXJpZXNHRik2JUYkL0kjeGlHRiUiIiEiIiVGNkYy

KydJI3hpRzYiLCQqJiwoKiZJI1Z8aXJHRiQiIiJJJ2RlbHRhfGlyR0YkRipGKkYpISIiRipGKkYqLCZGKEYqIiIjRipGLEYuRiotSSJPRyUqcHJvdGVjdGVkRzYjRioiIiQ=

QyY+SSZjb3NhMkc2Ii1JKXNpbXBsaWZ5RzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YlNiQqJiwmKiQmSSRzb2xHRiU2IyIiI0YyISIiIiIiRjRGNCwmRi5GNEY0RjRGM0kpc3ltYm9saWNHRiVGMy1GJzYkLUknc2VyaWVzR0YpNiVGJC9JI3hpR0YlIiIhIiInRjZGNA==

KydJI3hpRzYiIiIiIiIhLUkiT0clKnByb3RlY3RlZEc2I0YlIiIj

Final rezults: conserved quantities in O with sin/cos alpha given by Eqs. (32)/(33) QyQ+SSZFX3Jlekc2Ii1JKXNpbXBsaWZ5RzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YlNiQtSSVzdWJzR0YpNiUvLUkkc2luR0YoNiNJJmFscGhhR0YlSSZzaW5hMkdGJS8tSSRjb3NHRihGMkkmY29zYTJHRiVJIkVHRiVJKXN5bWJvbGljR0YlISIi QyQ+SSdQMV9yZXpHNiItSSlzaW1wbGlmeUc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJTYkLUklc3Vic0dGKTYlLy1JJHNpbkdGKDYjSSZhbHBoYUdGJUkmc2luYTJHRiUvLUkkY29zR0YoRjJJJmNvc2EyR0YlSSNQMUdGJUkpc3ltYm9saWNHRiUhIiI= QyQ+SSdQMl9yZXpHNiItSSlzaW1wbGlmeUc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJTYkLUklc3Vic0dGKTYlLy1JJHNpbkdGKDYjSSZhbHBoYUdGJUkmc2luYTJHRiUvLUkkY29zR0YoRjJJJmNvc2EyR0YlSSNQMkdGJUkpc3ltYm9saWNHRiUhIiI= QyQ+SSdRMV9yZXpHNiItSSlzaW1wbGlmeUc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJTYkLUklc3Vic0dGKTYlLy1JJHNpbkdGKDYjSSZhbHBoYUdGJUkmc2luYTJHRiUvLUkkY29zR0YoRjJJJmNvc2EyR0YlSSNRMUdGJUkpc3ltYm9saWNHRiUhIiI= QyQ+SSdRMl9yZXpHNiItSSlzaW1wbGlmeUc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJTYkLUklc3Vic0dGKTYlLy1JJHNpbkdGKDYjSSZhbHBoYUdGJUkmc2luYTJHRiUvLUkkY29zR0YoRjJJJmNvc2EyR0YlSSNRMkdGJUkpc3ltYm9saWNHRiUhIiI= Verify Eq. (59) QyQ+SSNQUEc2IiwmKiRJI1AxR0YlIiIjIiIiKiRJI1AyR0YlRilGKiEiIg== QyQ+SSxzaW5hX3NxX29ic0c2IiomSSNQMkdGJSIiI0kjUFBHRiUhIiJGKg== QyQ+SSROdW1HNiItSSZudW1lckclKnByb3RlY3RlZEc2I0ksc2luYV9zcV9vYnNHRiUhIiI= QyQ+SSREZW5HNiItSSZkZW5vbUclKnByb3RlY3RlZEc2I0ksc2luYV9zcV9vYnNHRiUhIiI= QyQ+SSVOdW0xRzYiLUklc3Vic0clKnByb3RlY3RlZEc2JS8tSSRzaW5HNiRGKEkoX3N5c2xpYkdGJTYjSSZhbHBoYUdGJUkmc2luYTJHRiUvLUkkY29zR0YtRi9JJmNvc2EyR0YlSSROdW1HRiUhIiI= QyQ+SSVEZW4xRzYiLUklc3Vic0clKnByb3RlY3RlZEc2JS8tSSRzaW5HNiRGKEkoX3N5c2xpYkdGJTYjSSZhbHBoYUdGJUkmc2luYTJHRiUvLUkkY29zR0YtRi9JJmNvc2EyR0YlSSREZW5HRiUhIiI= QyQ+SSZSZXo1OUc2IiooLUkiKkclKnByb3RlY3RlZEc2JCIiJSwmKiRJIlZHRiUiIiMhIiIiIiJGMUYxSSN4aUdGJUYvLCgqKEYuRjFJJmRlbHRhR0YlRjFGMkYvRjEqJkYuRjFGNUYxRjAhIiNGMUY3RjE=

LCQqKCwmKiRJI1Z8aXJHNiIiIiMhIiIiIiJGKkYqSSN4aUdGJ0YoLCgqKEYmRipJJ2RlbHRhfGlyR0YnRipGK0YoRioqJkYmRipGLkYqRikhIiNGKkYwIiIl

QyQtSSlzaW1wbGlmeUc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYkLCYqJkklTnVtMUdGKCIiIkklRGVuMUdGKCEiIkYtSSZSZXo1OUdGKEYvSSlzeW1ib2xpY0dGKEYt

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Expansions: sin(aplha) and Z=1/(1+z) QyQ+SSpFeHBfc2luYTJHNiItSSlzaW1wbGlmeUc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJTYkLUknc2VyaWVzR0YpNiVJJnNpbmEyR0YlL0kjeGlHRiUiIiEiIidJKXN5bWJvbGljR0YlIiIi

KylJI3hpRzYiLCQqJiwoKiZJI1Z8aXJHRiQiIiJJJ2RlbHRhfGlyR0YkRipGKkYpISIiRipGKkYqLCZGKEYqIiIjRipGLEYuRiosJCoqLDQqJkYpRi5GKyIiJEYqKiZGKUYuRitGLiEiJComRilGLkYrRioiIiUqJkYpRipGK0YuRjMqJEYpRi4hIiNGKCEiJ0YpRjdGK0YuRjpGKkYqRilGKiwoRjRGKkYoRjdGN0YqRixGLUYsRjpGMy1JIk9HJSpwcm90ZWN0ZWRHNiNGKiIiJg==

QyQ+SSZFeHBfWkc2Ii1JKXNpbXBsaWZ5RzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YlNiQtSSdzZXJpZXNHRik2JSomSSZFX3JlekdGJSIiIkkia0dGJSEiIi9JI3hpR0YlIiIhIiIpSSlzeW1ib2xpY0dGJUYx

KylJI3hpRzYiLCQqJiwsKiZJI1Z8aXJHRiQiIiJJJ2RlbHRhfGlyR0YkIiIjRioqJkYpRipGK0YqISIjRilGLEYrRixGLkYqRiosJiokRilGLCEiIkYqRiojRjFGLEYyIiIhLCQqKCw8KiZGKSIiJEYrIiIlRjgqJkYpRjhGK0Y4ISIpKiZGKUY4RitGLCIjNSomRilGLEYrRjgiIzcqJkYpRjhGK0YqRjsqJkYpRixGK0YsISNDKiRGKUY4RjkqJkYpRixGK0YqIiM/RigiIzlGMEY7Ri0hIztGKUY5RitGOUYqRi9GMiwoRkFGKkYtRjlGOUYqRjEjRipGLEYsLUkiT0clKnByb3RlY3RlZEc2I0YqRjk=

JSFH